On the rate of convergence of series of random variables

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ON THE RATE OF CONVERGENCE OF SERIES
OF RANDOM VARIABLES








By

EUNWOO NAM


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1992














To Sylvia, Petra, and Daniel















ACKNOWLEDGEMENTS


First of all, I would like to express my deep gratitude to Dr. Andrew Rosalsky,

my dissertation adviser, for his guidance, advice, understanding, encouragement

and friendship. I would like to thank Dr. Rocco Ballerini, Dr. Malay Ghosh, Dr.

Richard Scheaffer and Dr. Murali Rao for serving on my dissertation committee.

Also, I would like to thank Dr. Ronald Randles, Chairman of the Department of

Statistics, for his support and encouragement through my years at the University

of Florida.

In addition, let me express my appreciation to the Korean Air Force Academy

and Korean Air Force for their support of my studies at this university.

Most importantly, I wish to express my special thanks to my family, especially

my wife, Sooyeon, for her love, patience, support, and unceasing prayers, and my

children, Hwajung and Wontae, for being a joy to us. I am grateful to our parents

for their teaching me the principle of life.

Finally, I would like to thank my colleagues and friends for their assistance and

continuous prayers.















TABLE OF CONTENTS


page

ACKNOWLEDGEMENTS .................................................. iii

ABSTRACT............... .......................................... v

CHAPTER


1 INTRODUCTION ........................................... 1

2 TAIL SERIES STRONG LAWS OF LARGE NUMBERS I ...... 9

2.1 Introduction and Preliminaries ......................... 9
2.2 Tail series SLLNs for Arbitrary Random Variables ...... 14
2.3 Tail series SLLNs for Independent Random Variables ... 23
2.4 Examples ............................................. 31

3 TAIL SERIES WEAK LAWS OF LARGE NUMBERS ......... 42

3.1 Introductory Comments, Tail Series Inequality,
and a New Proof of Klesov's Tail Series SLLN ....... 42
3.2 Tail Series WLLNs .................................... 48

4 TAIL SERIES STRONG LAWS OF LARGE NUMBERS II .... 55

4.1 Introduction and Preliminaries ....................... 55
4.2 Tail series SLLNs ..................................... 57
4.3 The Weighted I.I.D. Case ................................ 73

5 SOME FUTURE RESEARCH PROBLEMS .................. 81


REFERENCES..................... ......................................

BIOGRAPHICAL SKETCH..............................................















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ON THE RATE OF CONVERGENCE OF SERIES
OF RANDOM VARIABLES

By

Eunwoo Nam

December 1992


Chairman: A. Rosalsky
Major Department: Statistics

For an almost surely (a.s.) convergent series of random variables S, = = l Xj,

the tail series T, = Ej=, Xj is a well-defined sequence of random variables which

converges to 0 a.s. The rate of a.s. convergence of S, to a random variable S is

investigated through the current study of the rate in which T,, converges to 0 a.s.

Tail series strong laws of large numbers (SLLN) of the form blT,, -- 0 a.s. (where

{b,, n > 1} is a sequence of positive constants with b,, 0) are obtained under

various sets of conditions. Both the cases of (i) {X,, n > 1} having no conditions

on their joint distributions and (ii) {X,, n > 1} being independent are investigated.

Some earlier work by Klesov on the tail series SLLN problem, which had provided

tail series analogues of Petrov's SLLNs for partial sums, is generalized to a larger

class of random variables. In the case of independent summands, some tail series








analogues of Teicher's SLLNs for partial sums are obtained as well.

Moreover, by employing the von Bahr and Esseen inequality, tail series weak laws

of large numbers (WLLN) for independent random variables are obtained. The tail

series WLLNs provide a bound on the rate in which supj>, ITjl converges to 0 in

probability. These tail series WLLNs are compared with the tail series SLLNs and

with tail series laws of the iterated logarithm of Rosalsky.

Examples are provided throughout to illustrate the current results and to com-

pare them with other results in the literature.














CHAPTER 1
INTRODUCTION


The theory of partial sums of random variables has been at the forefront of

research in statistical science for most of this century. The case of independent

summands has been of especial interest. One of the most interesting problems in

classical probability theory has been to determine, for a given series of random

variables, the probability that the series converges. (Here, and throughout the

entire sequel, the term "converges" means that the limit under consideration exists

and is finite. The term "diverges" means "does not converge.") According to the

famous Kolmogorov 0-1 law (see, e.g., Chow and Teicher [14], p. 64, or Chung [17],

p. 254), a series of independent random variables either converges almost surely

(a.s.) or diverges a.s. The primary objective of the current work is to determine

the almost sure rate of convergence for a convergent series. This objective will be

discussed in more detail below.

Let

Sn = EXj n> 1,
j=1
where {X,, n > 1} are random variables. This dissertation will concentrate on a

series of independent random variables, but some results are obtained without as-

suming independence. To establish almost sure convergence of the series Sn assum-

ing {X,, n > 1} are independent random variables, the Khintchine-Kolmogorov

1








convergence theorem (see, e.g., Chow and Teicher [14], p. 110) and the celebrated

Kolmogorov three-series criterion (see, e.g., Chow and Teicher [14], p. 114, or Chung

[17], p. 118) are very useful devices. In fact, the Kolmogorov three-series criterion

provides a triumvirate of conditions which are both necessary and sufficient for the

convergence of the series S, when the summands {X,, n > 1} are independent

random variables.

The Khintchine-Kolmogorov convergence theorem asserts that if {X,,, n > 1}

are independent random variables with

00
E(X,) = 0, n > 1 and f E(X)) < oo,
n=l

then the series S, converges a.s. and in quadratic mean to a random variable S

with

E(S) = 0 and E(S2) = f E(X ).
n=l

The Kolmogorov three-series criterion asserts that if {X,, n > 1} are independent

random variables, then the series S, converges a.s. iff


(i) PIX, > 1} < oo,
n=l

(ii) E E(X(x)) converges,
n=l

(iii) Var(X() < oo,
n=1

where


Xn") = Xnl4lxl<,], n > 1.






3

If the series Sn converges a.s. to a random variable S, then (set So = Xo = 0)

the tail series
oo
Tn = S Sn- = Xj, n n> 1
j=n
is a well-defined sequence of random variables and converges to 0 a.s. In the the-

ory of partial sums, the fact that the sum Sn is well defined for every n is, of

course, automatic. On the other hand, in the theory of tail series, the problem as

to whether {Tn, n > 1} is well defined is a genuine one. The two classical the-

orems (Khintchine-Kolmogorov convergence theorem and Kolmogorov three-series

criterion) play a key role in guaranteeing that the tail series Tn is well defined in

the case of independent summands. In this dissertation, we will focus on the rate

of convergence of the series Sn to a random variable S or, equivalently, on the rate

of convergence of the tail series T, to 0.

We say that the sequence of random variables {X,, n > 1} (such that the series

Sn diverges a.s.) obeys the strong law of large numbers (SLLN) with norming

constants {an, n > 1} if

Sn
0 a.s.
a,

where {a,, n > 1} is a sequence of positive constants with an T oo.

In the same way, we will say that the sequence {X,, n > 1} obeys the tail series

SLLN with norming constants {b,, n > 1} if the tail series Tn is well defined and


--+ 0 a.s.
bn


where {b,, n > 1} is a sequence of positive constants with bn 1 0.








Of course, a SLLN has a sharper result for the slower 0 < a, f oo. Similarly, a

tail series SLLN has a sharper result for the faster 0 < b,. 0.

SLLNs for partial sums lie at the very foundation of statistical science and

have been and still are the subject of vigorous research activity. In the case of

partial sums, the SLLN problem was investigated prior to the LIL problem. On the

other hand, the situation is reversed for tail series; the tail series LIL problem was

investigated prior to the tail series SLLN problem. As will be seen, many results

for partial sums S, can be paired with analogous results for tail series T,. (Of

course, the actual random variables in the series S, and tail series T, are necessarily

different.) This duality was first discovered and investigated by Chow and Teicher

[13].

Chow and Teicher [13] constructed a milestone for research about the rate of

almost sure convergence of the tail series of independent random variables. They

developed a tail series counterpart to the renowned Kolmogorov law of the iterated

logarithm (LIL) (see, e.g., Chow and Teicher [14], p. 343, or Petrov [40], p. 292) for

series of independent and bounded random variables. Studies to eliminate Chow

and Teicher's boundedness assumption were conducted by Barbour [9], Heyde [23],

Budianu [12], Rosalsky [41], and Klesov [29]. Barbour [9] suggested a methodology

which yields a tail series analogue of the Linderberg-Feller version of the central limit

theorem (CLT) (see, e.g., Chow and Teicher [14], p. 291, or Lorve [34], p. 292).

Using this methodology, Heyde [23] obtained tail series analogues of the CLT and

LIL for a martingale difference sequence. Budianu [12] proved a tail series LIL for








series of independent unbounded random variables, which is a tail series analogue

of Petrov [40, Section 10.2, Theorems 2] LIL, and extended it for two-dimensional

random variables. Rosalsky [41] developed a more general tail series LIL than that

of Budianu [12] for series of independent unbounded random variables, which is a

tail series counterpart to Teicher's [45] version of the LIL, and as special cases he

proved tail series LILs for weighted sums of independent and identically distributed

(i.i.d.) random variables (see below for the definition of the weighted i.i.d. case).

In the same year as Rosalsky's article appeared, Klesov [29] developed a version of

the tail series LIL for weighted i.i.d. unbounded random variables, but Klesov's [29,

Proposition 4] result is nothing but the special case / = 0 of Rosalsky [41, Theorem

2]. Klesov [29] also proved two tail series SLLNs for independent random variables,

which are tail series analogues of Petrov [40, Section 9.3, Theorems 12 and 13] and

Petrov [38, Theorem 5], respectively.

In his follow-up article, Klesov [30] extended his previous tail series SLLNs to

wider classes of independent random variables, and he also obtained tail series

SLLNs for several other dependence structures, viz. arbitrary sequences with no

assumptions on their joint distributions, orthogonal sequences, quasistationary se-

quences, and martingale difference sequences.

Solntsev [43] proved a tail series SLLN, but his result is not satisfactory since

his condition involves blocks


E Xi, k >l
j=n--l+l

of summands rather than individual summands. It is widely discussed in the








literature (see, e.g., Chung [16] or Lobve [34], p. 270) that conditions for the classical

SLLN for partial sums which involve blocks of random variables (as opposed to only

the individual summands) are unsatisfactory or at best undesirable. Such criticism

carries over directly to the tail series situation.

Throughout the entire sequel, all random variables are defined on a fixed but

otherwise arbitrary probability space (fl, F, P), and the logarithm and iterated

logarithm are conveniently defined for x > 0 and a positive integer r, by


Slog if x > e
logic x =
x if x < e

and

log, = log, log,_. z, r > 2

where log z (when x > e) denotes the natural logarithm.

Some of the results herein as well as some examples illustrating them con-

cern the weighted i.i.d. case consisting of sequences {X,, n > 1} of the form

X, = anY, 2> 1, where {Yn, n > 1} are i.i.d. random variables with E(Yi) =

0, E (Y2) = 1, and {an, n > 1} are nonzero constants.

This dissertation will be divided into five chapters which will now be briefly

described. In Chapter 2, we will generalize some of Klesov's [30] tail series SLLNs.

(Throughout this chapter and the subsequent ones, our assumptions on the ran-

dom variables {Xn, n > 1} involve the individual summands rather than blocks

of summands as were considered by Solntsev [43].) Furthermore, we will develop

truncated versions of our new tail series SLLNs. Both of the cases








(i) {X,, n > 1} are independent random variables,

and

(ii) {X,, n > 1} are random variables with no assumptions being imposed

on their joint distributions

are investigated. Also we will provide examples which demonstrate that the new

results are indeed better than previous ones.

Chapter 3 is quite independent of the others. In Chapter 3, we will study the

rate of convergence in probability of a series S, of independent random variables to

a random variable S or, more specifically, the rate in which supi>n ITjl converges

to 0 in probability by establishing tail series weak laws of large numbers (WLLN).

These tail series WLLNs take the form

sup ITj
j__ P
-----} 0
bn

where {b,, n > 1} is a sequence of norming constants with 0 < b, 1 0. As special

cases of our tail series WLLNs, we will obtain the tail series WLLNs for weighted

sums of i.i.d. random variables. Also, via the example of the harmonic series with

a random choice of signs, we will find a sequence of norming constants which yields

tail series WLLNs, but does not yield tail series SLLNs.

In Chapter 4, we will prove advanced tail series SLLNs for series of independent

random variables, which are counterparts to Teicher's [47] SLLNs for partial sums.

As special cases of these tail series SLLN'S, we will investigate the tail series SLLN

problem for weighted sums of i.i.d. random variables. Also we will provide an

example which illustrates the new results.






8

Finally, in the last chapter (Chapter 5), some problems for future research work

are presented.














CHAPTER 2
TAIL SERIES STRONG LAWS OF LARGE NUMBERS I


2.1 Introduction and Preliminaries


The tail series LIL has comparatively rich references; on the other hand, the

tail series SLLN has limited references. Two papers of Klesov [29 and 30] are

good references of the tail series SLLN. Although Klesov [29] proved two tail series

SLLNs for independent random variables by first establishing a tail series version

of the Kolmogorov's inequality (see, e.g., Chow and Teicher [14], p. 127 or Petrov

[40], p. 52), the proof of his tail series version of the Kolmogorov's inequality

is very complicated, and we could not quite follow his argument. His argument

rested on a tail series inequality which he did not substantiate. But, in his follow-

up article, using a tail series analogue of the Kronecker lemma (rather than his tail

series version of the Kolmogorov's inequality), Klesov [30] extended his previous tail

series SLLNs to wider classes of independent random variables. He also developed

tail series SLLNs for arbitrary random variables (i.e., not necessarily independent)

as well.

Some of Klesov's [29, 30] work will now be described. Let be the class of

functions O(z) satisfying the following three conditions:


(i) Ok(z) is positive and nondecreasing.








1
(ii) x ( -) tends monotonically to 0 as x 1 0.
x

(iii) E 1 <

Examples of such functions O(x) are

(xa) = IZl, 0 < < 1


W() = (log1 jl)1', e > 0

O() = (log 1zl|)(log, Il)'+e, e > 0

and so on.

For arbitrary random variables {Xn, n > 1}, without the assumption of inde-

pendence, Klesov [30] developed two tail series SLLNs (Propositions 1 and 2 below).

But Proposition 2 included a technical error in its formulation, and so it needs to

be restated. As in Chapter 1, {T,, n > 1} denotes throughout the sequence of tail

series T, = Ejj Xj, n > 1, corresponding to random variables {X,, n > 1}. Note

that the hypotheses of Proposition 1 and 2 ensure that {T,, n > 1} is well defined.

Proposition 1 (Klesov [30]). Let 0 < p < 1 and let {X,, n > 1} be random

variables. Furthermore, let {bn, n > 1} be a sequence of positive constants with

bn 0. If the series

SE(IXn)
< oo,
n=1
then the tail series SLLN
Tn
--- 0 a.s.
b,


obtains.








Proposition 2 (Klesov [30]). Let 0 < p < 1 and let {X,, n > 1} be random

variables. If the series

E E(IXIP) < oo,
n=1
then setting

An = E(IXiP), n> 1,
j=n
and assuming that An > 0, n > 1, the tail series SLLN

T,
-_ -+ 0 a.s.
(At(A;1))
obtains for each function O(x) E T.

Proposition 2 is a tail series analogue of a SLLN of Petrov [39].

Under the assumption that {X,, n > 1} are independent random variables,

Propositions 1 and 2 have been extended by Klesov [30] to Propositions 3 and 4

below, respectively, by employing a class of functions instead of a specific function

g(x) = IxzI, 0 < p < 1. But both of them included a technical error in their

formulation, and so they need to be reformulated as follows. In addition, Klesov

[30] did not verify that his conditions ensure that the tail series {T, n > 1} is

indeed well defined.

Let the function g(x) be positive for z > 0 with g(z) T oo as z T oo. Assume

that either of the following two conditions holds


(i) -- is nondecreasing for x > 0.
g(x)

(ii) g() is nondecreasing for x > 0, -- is nondecreasing for x > 0,
x g;(x)


and E(X) = 0, n > 1.






12

Proposition 3 (Klesov [30]). Let {X, n > 1} be independent random variables

and let {bn, n > 1} be a sequence of positive constants with bn i 0. If the series

0 E(g(|X,|))
o < oo,
n=1 g(bn)

then the tail series SLLN
Tn
0 a.s.
bn
obtains.

Proposition 4 (Klesov [30]). Let {Xn, n > 1} be independent random variables.

If the series

EE(g(IXn1)) n=l
then setting
0o
An = E(g(IXiI)), n>
j=n
and assuming that An > 0, n > 1, the tail series SLLN

Tn
7 0 a.s.
g-1 (An (A-1))

obtains for each function (zx) E T.

Not only do Propositions 3 and 4 reduce to two tail series SLLNs of Klesov

[29], respectively, by taking g(x) = xIP, 0 < p < 2, but they also are tail series

analogues of Petrov [40, Section 9. 3, Theorem 11 with gn = g, n > 1] and Petrov

[38, Theorem 5], respectively.

Most of our results in this chapter are based on the following two lemmas.

Lemma 1 (Heyde [23], Rosalsky [41], Klesov [30]). Let {z,, n > 1} be a sequence

of constants and let {b., n > 1} be a sequence of positive constants with bn 0. If








the series

converges,
n=1 bn

then
100
E j 0o.
-- j=n



Lemma 1 is a tail series analogue of the Kronecker lemma. This lemma is initially

due to Heyde [23], but Rosalsky [41] re-proved it in an alternative way because

Heyde's original proof was not clear. One year after Rosalsky's [41] paper appeared,

but independently from Rosalsky's paper, Klesov [30] proved the lemma in a manner

similar to that of Rosalsky. As we mentioned earlier, in his previous paper, Klesov

[29] proved his tail series SLLNs via a tail series version of the Kolmogorov inequality

instead of the above tail series analogue of the Kronecker lemma. The approach

using this analogue of the Kronecker lemma is simpler and indeed more natural.

Lemma 2 (Klesov [29]). Let {cn, n > 1} be a sequence of nonnegative constants

such that E, 1 c, < oo. If


C, = cj > 0, n>l,
j=n

then
00
C c < 00oo
n=1 C.n (C,;)

obtains for each function tk(z) E T.

This lemma is a tail series analogue of the Abel-Dini theorem (see, e.g., Knopp

[32], p. 290).








2.2 Tail series SLLNs for Arbitrary Random Variables


For arbitrary random variables {X,, n > 1}, without the assumption of inde-

pendence, we obtain the following tail series SLLNs. To avoid trivial considerations,

assume that {X,, n > 1} are not eventually degenerate at 0. This assumption is

in effect throughout the entire chapter and will not be repeated. The main result

of this section, Theorem 1, may now be stated. It will be shown in the proof of

Theorem 1 that the hypotheses ensure that {T,, n > 1} is a well-defined sequence

of random variables. The proof of Theorem 1 will be deferred until after the proof

of the ensuing Lemma 4.

Theorem 1. Let {Xn, n > 1} be random variables and let {g,(x), n > 1} be

strictly increasing functions defined on [0, oo) such that


gn(0) = 0 and lim gn(z) = oo, n > 1. (2.2.1)


Assume that

x
-g ) is nondecreasing as 0 < x T for each n > 1 (2.2.2)
9n(z)

and

gn(z) is nondecreasing in n for each fized z > 0. (2.2.3)

If the series
00
E E(gs(lXn|)) < oo, (2.2.4)
n=l

then setting

A= E(gi,(IXj)), n>
j=n







and assuming that for some function O(z) E T


P {(Xn < g;1 (An -k(An1)) eventually } = 1, (2.2.5)

the tail series SLLN
T,
---,- -- 0 a.s. (2.2.6)
9g(Ans (A-1))
obtains, where g;1 denotes the inverse function of g, for each n > 1.

Remarks. (i) By the Borel-Cantelli lemma, a sufficient condition for (2.2.5) is


EP {Ix > g(An(A ))}<00oo.
n=l

(ii) From the definitions of A, and the class %, we note that An (An1) 1 and

so g;-'(A,, (A-1)) 1. Moreover, the condition (2.2.5) is necessary for (2.2.6) to

hold. This follows from the remark after the ensuing Lemma 4 by setting b, =

g(.An ( 1)), ~ > 1.

(iii) For each n > 1, note that (2.2.2) together with the fact that each g,,() is a

nondecreasing function, implies that each g,(x) is necessarily a continuous function.

In order to prove this, we will show that


g,(xo) = ,,(x+) for arbitrary xo E (0, oo) and for each n > 1. (2.2.7)

Let 0 < s < xo < t. Then (2.2.2) ensures that

s zo t
< < n 1.
gn(s) 9n(xzo) gn(t)'

Take s T xo and t I xo. Then

=l m s t no
X lim < lm = X n 1.
7n(xo) -T0 9g(s) tIx 9gn(t) g.(Z+)








Therefore, g,(xo) > g9,(X) for arbitrary xo E (0, oo) and for each n > 1. Hence,

via the monotonicity of each gn, (2.2.7) follows. O

Assuming (2.2.5) (which is necessary for (2.2.6)), then not only does Theorem

1 reduce to Proposition 2 by setting


gn(x) =I, 0

1,


but, Theorem 1 also yields Theorem 2 under the condition (i), without the inde-

pendence assumption.

The proof of Theorem 1 utilizes the following two lemmas.

Lemma 3. Let {X,, n > 1} be random variables and let {gn(z), n > 1} be non-

decreasing functions defined on [0, oo) satisfying (2.2.1) and (2.2.2). Furthermore,

let {bn, n > 1} be a sequence of positive constants such that


P {IXn < bn eventually } = 1. (2.2.8)

If the series

oo (bgn( < oo, (2.2.9)
n=1 g,(bn)
then the series

y converges a.s. (2.2.10)
n=1 bn



Remark. Since (2.2.10) ensures that b.;X, -- 0 a.s., (2.2.8) follows. Thus the

condition (2.2.8) is necessary for (2.2.10) to hold.

Proof of Lemma 3. By (2.2.8), for almost all w E fl there exists an integer N(w)





17

such that IXn(w)l < b, for all n > N(w), and hence by (2.2.2)

IX,(w) < b.
Xn(XW)) < (bn), n > N(w). (2.2.11)
gn(jXn(W)j) g- n(b,)

Next, via the Lebesgue monotone convergence theorem, (2.2.9) ensures that


E ( ng.(bXn.) < 0

and so


Sg(bX) < oo a.s. (2.2.12)
E= gn(bn)

Thus, for almost all w E

SIX(W)I N() IX(w) IX-(w)I
n=-1 b ()+ E x (b ) (
n= n=N(w)+l
N'")IX.(w )I = 9.(X(w)l)
S. + ) (by (2.2.11))
1 b, n=N(w)+l g"(b")
< oo (by (2.2.12))

and therefore (2.2.10) obtains. o

Using Lemmas 1 and 3, we obtain the following lemma.

Lemma 4. Let {X,, n > 1} be random variables and let {9g(z), n > 1} be non-

decreasing functions defined on [0, oo) satisfying (2.2.1) and (2.2.2). Let {b,, n > 1}

be a sequence of positive constants satisfying (2.2.8) with b, I 0. If gn(bn) = 0(1)

and (2.2.9) holds, then the tail series SLLN

Tn
T --, 0 a.s. (2.2.13)
b,


obtains.








Remark. The triangle inequality and the fact that b, I imply

IX < IT.I IT.+1 ,
< + n_1
b. n b,+l '

and so (2.2.13) ensures b'X, -- 0 a.s. Thus the condition (2.2.8) is necessary for

(2.2.13) to hold.

Proof of Lemma 4. Note that (2.2.9) and g,(b,) = 0(1) ensure (2.2.4) which, as

will be demonstrated in the proof of Theorem 1, ensures that {T,, n > 1} is well

defined. Employing Lemma 3 yields (2.2.10). Since b, 1 0, the lemma follows from

Lemma 1. 0

By assuming (2.2.8) which is a necessary condition for (2.2.10) and (2.2.13),

Lemmas 3 and 4 yield the ensuing Lemmas 5 and 6, respectively, but without

assuming independence in the case when the condition (i) of Theorem 2 is assumed.

Also Lemma 4 reduces to Proposition 1 by setting


g,.(x) -= I, 0 < < 1, n > 1.

The proof of Theorem 1 may now be given.

Proof of Theorem 1. Firstly, we want to show that the tail series {T,, n > 1}

is well defined. Since, via the Lebesgue monotone convergence theorem, (2.2.4)

ensures

E g.( X1) < oo,
(n=1

(2.2.4) = g,(IX,|) < oo a.s. (2.2.14)
n=l
Sgn(lXn) -+ 0 a.s.

IX> X -+ 0 a.s. (by (2.2.3)). (2.2.15)








Now let N(w) be the random integer defined by


N(w) = min {N > 1 : IX,(w) I< 1 for all n > N} (= oo, otherwise ).


Then (2.2.15) ensures that N(w) < oo a.s. whence by (2.2.2), for almost all w E 0

IX., 1
(X-< -- n > N(w). (2.2.16)
9.(nxn -) 9n(l)

Thus, for almost all w E ft

oo N(w) oo
EIXnl = El X + E Ixnl
=n=1 =1 n=N(w)+l
N(w) oo0 lXl)
-< E |X + E (1X.|) (by (2.2.16))
n=1 n=N(w)+l gn(1)
N(w) 00 (1
E< IX.|+M E g,(IX.) M=
n=1 n=N(w)+l 91(1)
< oo (by (2.2.14))


and so


00
SX, converges a.s.
n=1
Therefore {T,, n > 1} is a well-defined sequence of random variables.

Next, let

cn = E(gn(jXn,)), n > 1

and observe that

00
c, > 0, n > 1 and E c, < oo a.s. (by (2.2.4)).
n=1

Since An = ZJl ci > 0, n > 1, Lemma 2 ensures that for each function tk(x) 6 l,

E E(n(lXnlj)) 0 E(n(lXn9 X))
S E(g.( ))(< oo.) ( 2.2.17)







For each function I(x) E *, since

o < g;1'(A O(A'1)) ) o,

then setting

b. = g (An (A;l)), n > 1,

(2.2.8) and (2.2.9) follows directly from (2.2.5) and (2.2.17), respectively. Thus the

theorem follows from Lemma 4 since g.(b,) = An b(A-1) = 0(1) (see Remark (ii)

after the statement of Theorem 1). 0

We obtain the following two truncated versions of Theorem 1 as corollaries.

Corollary 1. Let {Xn, n > 1} be random variables and let {g,(x), n > 1} be

strictly increasing functions defined on [0, oo) satisfying (2.2.1), (2.2.2) and (2.2.3).

If

SP {IIXnI > Cn} < oo (2.2.18)
n=1
and

E E(gn.(X. Illx.l n=l
are satisfied for some sequence of positive constants {Cn, n > 1}, then setting


= EE(g (IX, I jxil<,])), n 1
j=n
and assuming that for some function k(x) E '

P {IX, Ilx. lacn < n (: n b(.~)) eventually } = 1, (2.2.20)

the tail series SLLN
T.
-- T -+ 0 a.s. (2.2.21)
(^ nA. (;)) /








obtains.

Remarks. (i) A sufficient condition for (2.2.20) to hold is


EP {IIX. Illx.l g-(n -A:1 ))} < OO,
n=l

by the Borel-Cantelli lemma.

(ii) Since (2.2.18) asserts that {Xn, n > 1} and (X, I[xl 1} are equiv-

alent in the sense of Khintchine, (2.2.21) is equivalent to

T*
S--* 0 a.s. (2.2.22)


where T* Ejj Xj I[xj 1. Note that g,-(I.( (Anj 1)) 1. By the ar-

gument in the remark after the statement of Lemma 4, mutatis mutandis, (2.2.22)

ensures the condition (2.2.20). Thus the condition (2.2.20) is necessary for (2.2.21)

to hold.

(iii) The condition (2.2.18) ensures that (2.2.20) is equivalent to the apparently

stronger but structurally simpler condition


P {IX I < 9'1 (.An (a-1)) eventually } = 1.

Proof of Corollary 1. Set


Zn = Xn I[lx.l 1.

Then, by applying Theorem 1 to the random variables {Z,, n > 1}, (2.2.19) ensures

that the tail series T.* -= E Zj is well defined and then (2.2.22) obtains for each

function O(x) E C. Since {X,, n > 1} and {Xn I[ix.l 1} are equivalent in








the sense of Khintchine, {T., n > 1} is also well defined and the corollary follows.

0

Corollary 2. Let {X,, n > 1} be random variables and let {gn(x), n > 1} be

strictly increasing functions defined on [0, oo) satisfying (2.2.1), (2.2.2) and (2.2.3).

If (2.2.18) and

00
E E(g,(|X, I[ix. n=l

are satisfied for some sequence of positive constants {Cn, n > 1}, then setting


A= E(gj(lXj Il[x,l 1
j=n

and

00oo
Tn = E {X E(Xj I[Ixilc,])}, n > 1, (2.2.24)
j=n

and assuming for some function (b(x) E T that


P {IX I[x.l

the tail series SLLN

0 (2.2.26)


obtains.

Remarks. By the argument in Remarks (i), (ii), and (iii) after the statement of

Corollary 1, mutatis mutandis, we observe, respectively, that

(i) A sufficient condition for (2.2.25) is


oP {Xn I[x.l gn- (,n )) < 00.
n=1








(ii) The condition (2.2.25) is necessary for (2.2.26) to hold.

(iii) The condition (2.2.18) ensures that (2.2.25) is equivalent to the condition

P {IX, E(Xn I[Ixl
Proof of Corollary 2. Set

Z, = X, Il[x.nl 1.

Then the result follows from (2.2.18) and (2.2.23) by employing the argument in

Corollary 1, mutatis mutandis. o


2.3 Tail Series SLLNs for Independent Random Variables


For independent random variables {X,, n > 1} we obtain the following tail

series SLLNs. In part (i) of the ensuing theorem, the condition (2.2.5) of Theorem

1 is dispensed with at the expense of assuming that {X,, n > 1} are independent.

The main result of this section, Theorem 2, may now be stated. As in Theorem

1, it will be shown in the proof of Theorem 2 that the hypotheses ensure that

{ T, n > 1} is a well-defined sequence of random variables. The proof of Theorem

2 will be deferred until after the proof of the ensuing Lemma 6.

Theorem 2. Let {Xn, n > 1} be independent random variables and let {gn(z), n >

1} be strictly increasing functions defined on [0, oo) such that

g.(0) = 0 and limg(z) = oo, n > 1 (2.3.1)

and assume that


g,(z) is nondecreasing in n for each fixed z > 0.


(2.3.2)








Suppose that one of the following two conditions prevails

(i) is nondecreasing as 0 < x T for each n > 1.
sn(xX2
gn (s))
(ii) gn(x) is nondecreasing as < is nondecreasing as 0 < x ,
x gn(x)

and E(X,) = 0, for each n > 1.

If the series
00
SE(g.(IX.I)) < oo, (2.3.3)
n=1
then setting

.= EE(gj(|IXj)), n> 1,
j=n
the tail series SLLN
Tn
----- 0 a.s.
g;1 ( An (A; ))
obtains for each function Ob(x) E 9, where g'1 denotes the inverse function of gn for

each n > 1.

Remark. Note that for each n > 1, the hypotheses to (i ) or (ii ), together

with the fact that each gn(z) is a nondecreasing function, imply that each g,(x) is

necessarily a continuous function. Under the hypotheses to (i ) the continuity of

each gn(s) follows directly from Remark (iii) after the statement of Theorem 1. So

it is enough to show that each gn(x) is a continuous function under the hypotheses

to (ii ). To this end, we will prove that

g,(xo) = g,(x+) for arbitrary xo E (0, oo) and for each n > 1. (2.3.4)

Let 0 < s < xo < t. Then (ii) of the theorem ensures that

sa2 T2 t2
< o < n > 1.
9n(s) gn(xo) gn(t)








Take s f xo and t I xo. Then

T2 s2 t2 2
= lim lilim 0 n>l
g.(zo) oe g-(s) t=o g,(t) g.f(zo+)

Therefore for each n > 1,


gn(zo) 2 Sg,(+) for arbitrary xo E (0, oo) and for each n > 1.

Hence, via the monotonicity of each gn, (2.3.4) follows. O

Theorem 2 reduces to Proposition 4 by setting g, = g, n > 1. And also Theorem

2, under the hypotheses to (i), follows directly from Theorem 1 by assuming the

condition (2.2.5) which is a necessary condition for the result to hold. Moreover, as

will become apparent, Theorem 2, under the hypotheses to (ii), owes much to the

work of Klesov [30].

The proof of Theorem 2 utilizes the following two lemmas.

Lemma 5 (Petrov [38]). Let {X,, n > 1} be independent random variables and

let {9g(x), n > 1} be nondecreasing functions defined on [0, oo) satisfying (2.3.1).

Assume that condition (i) or (ii) of Theorem 2 holds. Further, let {b., n > 1} be

a sequence of positive constants. If the series

SE(gn(,X < oo (2.3.5)
n=l gn(b.n)

then the series
0X
converges a.s. (2.3.6)
n=1 b"


Lemma 5, under the condition (ii) of Theorem 2, was proved for the case g,

g, n > 1, by Chung [17, p. 124].








Using Lemmas 1 and 5, we obtain the following lemma.

Lemma 6. Let {X,, n > 1} be independent random variables and let {g,n(), n >

1} be nondecreasing functions defined on [0,oo) satisfying (2.3.1). Assume that

condition (i) or (ii) of Theorem 2 holds. Let {b,, n > 1} be a sequence of positive

constants with b, 1 0. If g,(b) = 0(1) and (2.3.5) holds, then the tail series SLLN

Tn
-- 0 a.s.


obtains.

Proof. Note that (2.3.5) and g,(bn) = 0(1) ensure (2.3.3) which, as will be

demonstrated in the proof of Theorem 2, ensures that {Tn, n > 1} is well defined.

Employing Lemma 5 yields (2.3.6). Since b, 0, the lemma follows from Lemma 1.



Not only does Lemma 6 reduce to Proposition 3 by taking g, g, n > 1, but

it also is a tail series analogue of Petrov [40, Section 9. 3, Theorem 11]. Moreover,

if (2.2.8) holds, then under the hypotheses to (i) of Theorem 2, Lemmas 5 and 6

follow directly from Lemmas 3 and 4, respectively.

The proof of Theorem 2 may now be given.

Proof of Theorem 2. Note at the outset that in the proof of Theorem 1 the

condition (2.2.5) was not employed to establish that the tail series {Tn, n > 1} is

well defined. Consequently, under the hypotheses to (i), {T,,, n > 1} is a well-defined

sequence of random variables.

Next, it will be verified by employing the Kolmogorov three-series criterion that

under the hypotheses to (ii), E'= X, converges a.s. and hence {T,, n > 1} is well








defined. For each n > 1


P{IX, > 1} = P{g,(lXn,) > g(1)}
E(g,(|Xn|))
< E(g(X ) (by the Markov inequality)
g9 (1)


< ME(gn(IX|n)) (M=


9i1)


and so


oo
P {IXl|
n=-1


< oo (by (2.3.3)).


oo
> 1} < MEE(g(| XI))
n=1


Now for each n > 1


E (X, I[x.l

= IE(X, Ilx,l>]) I (since E(Xn) = 0)


x
g.n() -


1
9,'(1)'


a>l)


E(IXI I|lxl>])
E(gn(lX,I) Illxnl>]) (since
gn(1)
SE9n(lXnl))
n(l)


< ME(gn(|Xn)) (M


00
< M E E(g(IXI)) < oo
n=1


(by (2.3.3))


implying that


00
E(Xn I[|Ix.<1]) converges.
n=l


Again, for each n > 1


< E(X. I[Ix.I<])
E(gn(IX,I) I[ix,.1 Sg,(1) (since


x2
;gn(a)


1
< --- x<
9ng(1)


Thus,


91(1)


00
, EXn I[|x.1 n=l


Var(X, I[ix,.1<])








< E(g(|IXnj))
g- (1)
< ME(g(|IXI|)) (M


gi(l)


implying


00oo(2.3.3)).
< M E E(g9(IX.|)) < oo (by (2.3.3)).
n=1


Hence the conditions of the the Kolmogorov three-series criterion are satisfied

thereby ensuring that
00
E X, converges a.s.
n=l
Therefore, in both cases, {T,, n > 1} is a well-defined sequence of random variables.

Next, let


Cn = E(g9(||Xn)), n > 1


and observe that

00
c, > 0, n > 1, and E c, < oo (by (2.3.3)).
n=l

Since An = Ct=n c > 0, n 2 1, Lemma 2 ensures that for each function k(x) 6 T,


0 E(gn(lXnl))


oo E( ,.(|X ))
E /k ( ,)
nt=1ln(.'


Now for each function 'b(z) E I9, since


0 < g;1-((' O(A;')) 1 0,


setting


E Var(X, Igx.1])
n=l


bn = g (, 1(A )), n > 1,








the condition (2.3.5) holds. The theorem then follows directly from Lemma 6 since

g.(b.) = .A4 (A41) = 0(1). O

We obtain the following two truncated versions of Theorem 2 as corollaries by

employing an argument similar to that used to establish Corollaries 1 and 2 of

Section 2.1.

Corollary 3. Let {X., n > 1} be independent random variables and let {g,(z), n >

1} be strictly increasing functions defined on [0,oo) satisfying (2.3.1) and (2.3.2).

Assume that the condition (i) or (ii) of Theorem 2 holds. If

00
P {|IXI > C,} < oo (2.3.7)
n=1

and

YE E(g(lXn Iix.l n=1

are satisfied for some sequence of positive constants {C,, n > 1}, then setting

00oo
n = E E(g,(lXj I[Ix, 1,
j=n

the tail series SLLN

S--+ 0 a.s. (2.3.9)

obtains for each function b(zx) E x1.

Remark. A necessary condition for (2.3.9) to hold is that (2.3.7) obtains with


C, = g (An 1)), n > 1. (2.3.10)


Proof of Remark. The triangle inequality and the fact that g-' (A,n b(,1))







imply

Ix.I IT.nl IT.+l
< + n>1.
g-1 (A b(A;1)) 1 (An 0(A;')) g+ 1 (4 n .A41)) 1

Thus (2.3.9) ensures
X,
X 0 a.s.
g;1 (An A ))
Using the Borel-Cantelli lemma and the independence of {Xn, n > 1}, we obtain


P {IX l > g<(An < 0))< .
n=l

Hence (2.3.7) holds with {Cn, n > 1} as in (2.3.10). 0

Proof of Corollary 3. Set


Z, = X, I[IXl 1.

Then, by applying Theorem 2 to the random variables {Z,, n > 1}, (2.3.8) implies

that the tail series T, = Ej', Z, is well defined and the tail series SLLN

T*
--+ 0 a.s.
g; (An )

obtains for each function O(z)E % Since (2.3.7) implies that {X,, n > 1} and

{X, I[ix.i 1} are equivalent in the sense of Khintchine, {T,, n > 1} is also

well defined and Corollary 3 follows. O

Corollary 4. Let {X,, n > 1} be independent random variables and let {g.(z), n >

1} be strictly increasing functions defined on [0,oo) satisfying (2.3.1) and (2.3.2).

Assume that condition (i) or (ii) of Theorem 2 holds. If (2.3.7) and


SE(g,(IXX. I[llx.lc.] E(X, I(Ix._<:c.])|)) < 00 (2.3.11)
n=1








are satisfied for some sequence of positive constants {C., n > 1}, then setting

oo
A= E E(gj(IXj Ilx, c,] E(Xj I[IX, C])|)), n > 1
j=n

and

00oo
t= E {Xj -E(X lI[Ix, I1,
j=n

the tail series SLLN

-- 0 a.s.


obtains for each function 0b(x) E T.

Proof. Set

Z, = X. I[lx.l .] E(Xn I[lx.jc.,]), n > 1.


Then the corollary follows from (2.3.7) and (2.3.11) by employing the argument in

Corollary 3, mutatis mutandis. o


2.4 Examples


Three examples are provided to illustrate some of the current results as well as

to compare them with related results in the literature.

Example 1. Let {X,, n > 1} be random variables (not necessarily independent)

such that

PX, 1-- and P{,,e"= n>1.
n2 -

Then for any pE (0, 1],

1 ( 1 ) e"p
E(Xnl-) 1 + n >l
2p 2- n








and so

E E(IX./P) = oo for all p (0,1].
n=l
Therefore the hypotheses of Proposition 2 are not met.

Let < a < 1 and let
2


(2.4.1)


Then, recalling the definition of log1 x, the conditions (2.2.1), (2.2.2), and (2.2.3)

are satisfied. For each n > 1


E(gn,(Xn|)) = E((log, IX||)) = 1


1
n2a


E E(g (IX||))
n=1


+ na-2 < .


Hence (2.2.4) holds.

Now for n > 1,

00
A = ZE (log1 lXjl)")
j=n


Se 1 1ja
i=n le jl


(1 ) i -2}


M n-2a M n-1 (M = 1 1 and M2 =
M nha+M2l = ea 2a 1


h-a)


. (2.4.2)


Suppose that 1 < a < 2. Then


A) i tM nt1-2o


If O(x) is taken to be the function


(x) = (log,1 )1+" where e > 0,


and so


n"
n2


gn(x) =- (logz x0)", n > 1.


(2.4.3)








then for all large n


A b(A, ) ~ M3 n1-2 (log, n)+e (s= 3 (2a )')

= o(1).

Therefore, for all large n


implying


9g1 (A. (A-1)) = e (A (A-1)) M4 n-2 (log1 n) 16


(M4 = e Ma).


Thus,


P{IXn > gl -(An O(An-))I = n for all large n


and so, via Remark (i) after the statement of Theorem 1, the condition (2.2.5) also

holds. Hence for each a E (, j, the tail series SLLN

T7
1-2 -+ 0 a.s.
n-a (log, n) +


obtains by Theorem 1, i.e.,


2-
n a
2 Tn -- 0 a.s.
(log, n) 1


On the other hand, suppose that < a < 1. Then recalling (2.4.2),

An, M na-1.

If O(z) be taken to be the function as in (2.4.3), then for all large n

An (A;') ~ Ms n'-1 (log, n)1' (Ms = (1 a)')


= o(1)


(2.4.4)


(2.4.5)


A. (A.4x) < 1








implying

A ((A.;1) < 1

for all large n and so


(A,1 (A;')) = e (A. (A1)) I~ Men-* (log1 n)'W (Me = e Ms) .

Thus, (2.4.4) holds and so, via Remark (i) after the statement of Theorem 1, the

condition (2.2.5) also holds. Hence for each a E (j, 1), the tail series SLLN

T.
-- --+ 0 a.s.
n (logl n)

obtains by Theorem 1, i.e.,
1.-1
-l n T' -- 0 a.s. (2.4.6)
(log1 n)
Next, it will now be demonstrated that Corollary 1 can be also applied. Let

S< p < 1 and let

gn(x) Ix, n > 1. (2.4.7)

Then the conditions (2.2.1), (2.2.2), and (2.2.3) are satisfied. Set

C,, 2, n > 1. (2.4.8)

Then

1
P{IX,, > C} = -, n > 1
n

implying (2.2.18). Also

E(gn(|XI[nxl.
= (1 1), n2
n2p n2








and so

E g.(X. I[x. n=l n=l

Thus the condition (2.2.19) holds. Now for n > 1


A = E(I XI[xllxcI)
j=n

= 1 --


~ M7n-2p M7 =2 1)


If (zx) be taken to be the function as in (2.4.3), then for all large n,


A 0,(A-1) ~ Ms n1-2p (log n)l+" (Ms = (2p 1)')

= o(1).


Hence for all large n,




implying


= &(An i))p ~ MAn-2 (log, n)' (M9 = M ).


Thus, (2.4.4) holds and so, via Remark (i) after the statement of Corollary 1, the

condition (2.2.20) also holds. Therefore, by Corollary 1 the tail series SLLN

T7
1-2 7 --+ 0 a.s.
nP (logI n) p

obtains, i.e.,
2-1
n P
i Tn -+ 0 a.s.
(log1n) P








Not only is this result sharper than (2.4.5) for p > a, a E (, ], but, also,

this result is sharper than (2.4.6) for p > aE (0,1). Hence, for pE (,1],

Corollary 1 gives us a better result than that which can be obtained by Theorem 1.

In conclusion, it may be noted that Theorem 2 and Corollary 3 can also be

applied to this example by setting {g,, n > 1} as in (2.4.1) or as in (2.4.7), respec-

tively, if {X,, n > 1} are assumed to be independent.

Example 2. Let {X,, n > 1} be random variables (not necessarily independent)

such that
1 1
P{X = = 1= 1 and P{X, = en} = n2 1

Then for any p E (0, 1],


n 2 '

and so

E E(IX.') = oo, for all pE (0, 1].
n=l
Therefore the hypotheses of Proposition 2 are not met.

It will now be demonstrated that Corollary 2 can be applied. Let


Then by the argument in Example 1, setting {gn, n > 1} as in (2.4.7) and {C,, n >

1} as in (2.4.8), the conditions (2.2.1), (2.2.2), (2.2.3), and (2.2.18) are satisfied.

Moreover, for each n > 1


E(gn(IX. I[lx.c<.] E(X, Iix.
= E(X, IIxc] 1 ) )
1 (1 12 1
np 2 n2 n2








implying


EE(g(I X I[ix-I n=1 n=l n2p


+ }<00.
n2 I


Hence (2.2.23) holds.

Next for n > 1,


00
= E(IXj I[Ix, j=n


,=n1 1 (

~ M1n -2p (Mi


(2p-1 J
2p- 1


If O(x) be taken to be the function as in (2.4.3), then for all large n,


-A 0(1)


SM2 nl-2' logici n)'+ (M2 = (2p 1)')


= o(I).


Hence, for all large n


and so

(An A )) = ( ( )) Ms n-2 (log, n)P (Ms = M2) .


Thus, for all large n


P {IX, I.x 9g' (An ( '))}

= P {j[IIxn2]- P{IX,I < 2}1 > g; (An0 (in-'))

< P {[IXn < 2] n [I[Ilx.l2] P{X, <5 2} > g;' (j,. 0(A-'))] } + P{IXn\ > 2}









= P [jX.j<2]n 1-1+2- >g-i(A .-)) +P{|X|>2}

= P{IXn1 > 2}
1
n2

and so by Remark (i) after the statement of Corollary 2 the condition (2.2.25) also

holds. Therefore, by Corollary 2, the tail series SLLN


1-2 --+ 0 a.s.
n- (login ) P

obtains for the tail series {TI, n > 1} defined as in (2.2.24), i.e.,

22-t
-+ Tn -+ 0 a.s.
(log, n) P

In conclusion, it may be noted that Corollary 4 can also be applied to this

example by setting {g,, n > 1} as in (2.4.7) in the case when {X,, n > 1} are

assumed to be independent.

In the following example, we will consider the rate of almost sure convergence

of the harmonic series with a random choice of signs.

Example 3. Let {X,, n > 1} be independent random variables such that


P X, =P x,=- = nn>l.


The series of partial sums S, = '=ji Xj, n > 1, can be interpreted as the harmonic

series with a random choice of signs. We will employ Theorem 2 to determine its

rate of convergence to a random variable.

Let 0 < a < 1 and let


g,(z) = n'--a, n 2 1, and g(x) = gi(x) = zx, > 0.








Then the conditions (2.3.1), (2.3.2), and (ii) of Theorem 2 are satisfied. For each

n>1,

E(g,(IX.1)) = n-(x+a) and E(9(|XI)) = n-2

implying (2.3.3) and (2.1.1), respectively. Therefore, all the hypotheses of Theorem

2 as well as all the hypotheses of Proposition 4 (with g(x) = g1(z) = x2) are

satisfied.

Now for n > 1,

A = EE(gj(lXjl)) = E j-(+) M, n- (M, = a-1) (2.4.9)
j=n j=n

and

A. = Z E(g(IXj)) = j-2 n-1 (2.4.10)
j=n j=n
If k(xz) be taken to be the function




then

An (An -1) M2 n- (M2 = a-)

and

A,, (A;1)~ n-2

implying, respectively,

gl(^.4(A; ,l)) ( M3n-+ (M =-)


and

-(An i(A-1)) n .








Thus, by applying Theorem 2 and Proposition 4, the tail series SLLNs

n2 Tn -- 0 a.s. (2.4.11)


and

nT Tn -- 0 a.s. (2.4.12)

obtain, respectively. Hence, recalling a < 1, (2.4.11) dominates (2.4.12). Therefore

Theorem 2 gives us a sharper result than that which can be obtained by Proposition

4.

Next, by taking t(x) to be the function as in (2.4.3) of Example 1, two relations

(2.4.9) and (2.4.10) yield the asymptotic relations


An (A1') ~ M4 n- (log, n)'+' (M4 = a')

and

An b(A-) ~ n-'(log, n)'+,

respectively. Thus,


n (A ;')) Ms n-" (log, n)+'P (M, = at)

and

g-'(An,(A-1)) ~ n- (log, n).

Hence, by either Theorem 2 or Proposition 4, the tail series SLLN


(logn) Tn 0 a.s. (2.4.13)
(log, n) 2

obtains for arbitrary e > 0. Therefore, there is no advantage of Theorem 2 over

Proposition 4 in this case.








Furthermore, let {Y., n > 1} be a sequence of i.i.d. random variables such that

1
P{Y, = 1} = P{Y, = -1} = -, n > 1.


Consider the weighted i.i.d. random variables


Xn = a, Yn, where a, = n-1, n > 1.


Then
00 001 1
= E = n
j=n j=n
and so


na
2 = o(t+1) and = O(1).

Therefore, by a tail series LIL of Rosalsky [41, Theorem 2] where / therein is chosen

to be 0, the tail series LIL


m sup n aj Yj = 1 a.s.
n- o(2 t log2 t-)-2)

obtains, i.e.,

lim sup 2 n T = v2 a.s. (2.4.14)
n-o (log, n)I

Hence, for arbitrary e > 0, the tail series SLLN
I
n2
g2 Tn -- 0 a.s.
(log, n)I+c

obtains. Thus this result of the tail series LIL of Rosalsky [41, Theorem 2] is sharper

than (2.4.13) as well as (2.4.11) and (2.4.12).

This example illustrates the gap between the conclusion of the tail series SLLN

(Theorem 2) and that of the tail series LIL of Rosalsky [41, Theorem 2]. Further

discussion about this will be given in Chapter 4.














CHAPTER 3
TAIL SERIES WEAK LAWS OF LARGE NUMBERS


3.1 Introductory Comments, Tail Series Inequality,
and a New Proof of Klesov's Tail Series SLLN


As was mentioned at the beginning of Chapter 2, in Klesov's [29, Lemma 1]

proof of a tail series version of Kolmogorov's inequality for independent random

variables, not only was his argument obscure, but, also, he employed a tail series in-

equality without proving it. After formulating and proving this tail series inequality

(Proposition 5 below), we will provide an alternative proof of the tail series SLLN

of Klesov [29, Proposition 1] (which is not based on the tail series version of the

Kolmogorov's inequality as was used by Klesov to prove his tail series SLLN). As a

direct application of this tail series inequality, we will establish tail series WLLNs

for the case of independent summands. Furthermore, as special cases of these tail

series WLLNs, we will also obtain tail series WLLNs for weighted sums of i.i.d.

random variables. As in Chapters 1 and 2, {Tn, n > 1} denotes throughout the tail

series T, = iO=, Xj, n > 1, corresponding to random variables {X,, n > 1}. As

will be seen, the hypotheses to each of the tail series results presented below ensure

that {Tn, n > 1} is a well-defined sequence of random variables. Hence Tn --+ 0 a.s.

or, equivalently,

sup Tl P 0.
j>n








As was mentioned in Chapter 1, these tail series WLLNs are of the form

sup IT,
>_n P
-_-- 0 (3.1.1)
bn

where {b,,, n > 1} is a suitable sequence of norming constants with 0 < b, i 0.

Of course, if the tail series SLLN

T.
T --+ 0 a.s.
bn

holds, then

]Ti IP
sup 0
j>n bj

whence via 0 < b4 1 0 the tail series WLLN (3.1.1) also obtains and it involves the

same sequence of norming constants.

This tail series inequality under discussion may now be formulated.

Proposition 5 (Klesov [29]). Let {X,, n > 1} be independent random variables

with E(IX,IP) < oo, n > 1, for some p > 0. Assume that one of the following two

conditions holds

(i)0
(ii) 1 < p < 2 and E(X,) = 0.

If

E E(IX,,I) < oo, (3.1.2)
n=l
then for every e > 0, the inequalities


P{sup ITrI > 6 P E(|XjlI), n 2 1
j>nobt e j=n

obtain where Cn (p) E (0,2] is a sequence of constants depending only on p.








The proof of Proposition 5, which will be given below, utilizes the following

Lemmas 7 and 8 and the proposition, under the assumption (ii ), is indeed a tail

series analogue of Lemma 7 which concerns partial sums of independent random

variables.

Lemma 7. Let S, = Z=X Xj, n > 1 where {X,, n > 1} are independent random

variables satisfying for some p E (1,2]


E(IXIP) < oo and E(Xn) = 0, n > 1.

Then for all e > 0, the inequalities


P( max ISl > e < E(IXI), n > 1

obtain.

Proof. Note at the outset that the hypotheses ensure that {Sn, .F,, n > 1} is

a martingale where 7, = e(Xi, X2, ..., X,), n > 1, and so {|ISn 7n, n > 1}

is a submartingale (see, e.g., Chow and Teicher [14], p. 232) since the function

p(t) = ItlP is convex. Then


P{max Si > e = P max Sj' > e,
l < E(ISnIP)

(by Doob's submartingale maximal inequality [18, p. 314])
2"
< E E(IX I)
j=1

by employing the von Bahr-Esseen [8] inequality. Thus the lemma follows. o

Lemma 8. Let {Xn, n > 1} be independent random variables satisfying for some








pE (1,2]

E(IXnI) < oo and E(Xn) = 0, n > 1.

For each n > 1 and 1 < k < n, let

S.j= EXi, k i=j
Then for all choices of n and k with 1 < k < n and for all e > 0, the inequality

max
Pf Jax S.J > 6} EE(IXi P)
j=k

obtains.

Proof. Fix n > 1 and 1 < k < n. Set

Sj = .+i-, 1 i=1
and note that

{Sn,:j=k, .., n}= {S :j=n+ l k, ..., 1}.

Then, applying Lemma 7 to the random variables {Xn, X,-1, ..., Xk}, it follows

that for e > 0,

P{max ISni > e = Pf max ISI > e
1k 2 n+l-k
< E E(IXn+-j I-)
j=1
2 "
-- j (E(IXiIP)

thereby proving the lemma. 0

Proof of Proposition 5. Let g,(x) IxI, 0 < p < 2, n > 1. Then, by the

argument in the proof of Theorem 2 of Chapter 2, (3.1.2) ensures that {T,, n > 1}

is a well-defined sequence of random variables.







Firstly, suppose that the assumption (i) holds. Then


Pjsup ITi|> < pf P Xyll > c
jn j=nCo

= P (E IXi ) > e'

< E (( Ixj l)' (by the Markov inequality)

lim E (EI jX)'P
SP N-*oo
(by the Lebesgue monotone convergence theorem)
1 N
< lim E |IXj' ( since a + bIP < laI+ IbI, 0 eP N--*oo (j=n
1"00
= -E E(IXjIP)
j=n

again by the Lebesgue monotone convergence theorem. Thus, the proposition fol-

lows under the assumption (i) with C,(p) = 1.

Next, Lemma 8 will be employed to prove Proposition 5 under the assumption

(ii). Note that for N > n > 1,


P max |Tj > e = P max lim > e
n M
= P max lim i X| > e

SaNf M m
= P lim max | Xi| > e
M--oo n

I lim max I '.x,|>e)
S M-fooni
< E liminfr I
M--oo max IE" XI>.









< liminf E Ir (by Fatou's lemma)
M-0ooN | J

= liminfP max XlB xil> e
M-*oo n
2M
liminf P max IZXI > e
M-+oo n
liminf- 2 E(lX, I-) (by Lemma 8)
M--oo CP. n
2 oo00
= E E(IXiIP).


Letting N -- oo yields


P sup IT > e = lim P max T, > e
ujT> N-*oo n 2"0
< -C E E(IX IP)
j=n

thereby proving the proposition under the assumption (ii) with Cn(p) = 2. O

Now, using Proposition 5, we will re-prove (in Proposition 6 below) the tail series

SLLN of Klesov [29, Proposition 1] which we had questioned earlier. Of course,

Proposition 6 is merely the special case g(x) = IxIP, 0 < p < 2, of Proposition 3 of

Chapter 2 as well as the special case gn(x) = IzxI, 0 < p < 2, n > 1, of Lemma 6

of Chapter 2 but an alternative proof may be of interest.

Proposition 6 (Klesov [29]). Let {X,, n > 1} be independent random variables

with E(IXnI) < oo, n > 1, for some p > 0. Assume that either of the condition (i)

or (ii) of Proposition 5 holds. Let {bn, n > 1} be a sequence of positive constants

with b, 1 0. If the series
SE(IX IP)
E < o, (3.1.3)
n=1 b







then the tail series SLLN

--+ 0 a.s.
bn
obtains.

Proof. By the proof of Proposition 5 with X, replaced by bn'Xn, n > 1,

{(~ n b. lXj, n > 1) is a well-defined sequence of random variables. Since b, i 0,

the proposition follows from Lemma 1 of Chapter 2. 0


3.2 Tail Series WLLNs


Using Proposition 5, we will prove tail series WLLNs of the form

sup ITA
j2, P
---+0
bn

where {b,, n > 1} is a sequence of norming constants with 0 < b, i 0. This, of

course, ensures that
T, p
~ 0.
bn
The following theorem is comparable with Proposition 6.

Theorem 3. Let {X,, n > 1} be independent random variables with E(IX,'I)) <

oo, n > 1, for some p > 0. Assume that either of the conditions (i) or (ii) of

Proposition 5 holds. Let {b,, n > 1} be a sequence of positive constants with b, 1 0.

If

E E(IXjI) 0, (3.2.1)
j=n
then the tail series WLLN

sup ITil
i> P
0 (3.2.2)
bn








obtains.

Remark. Note at the outset that Lemma 1 of Chapter 2 ensures that (3.1.3)

implies (3.2.1). Thus, while in Theorem 3 we obtain a weaker conclusion than that

of Proposition 6, we use a weaker assumption.

Proof of Theorem 3. Since (3.2.1) implies (3.1.2), taking gn(z) =_ Ix\z, 0 < p <

2, n > 1, we see that {Tn, n > 1} is a well-defined sequence of random variables

by the argument in the proof of Theorem 2 of Chapter 2. Alternatively, it may be

noted that (3.2.1) implies (3.1.2) whence {T,, n > 1} is well defined as was shown

in Proposition 5.

In Proposition 5, replace e by ebn for each n > 1. Then, for arbitrary e > 0
p Isup | I Tj( 1
bP ,, > p E(jXjl ) (0 < C.(p) < 2)

0 (by (3.2.1))


thereby proving (3.2.2). O

Corollary 5. Under the hypotheses to Theorem 3, the tail series WLLN

Tn P
0
b.

obtains.

Proof. The corollary follows immediately from (3.2.2). 0

As additional corollaries of this theorem, we obtain the following two tail series

WLLNs (Corollaries 6 and 7) for the weighted i.i.d. case.

Corollary 6. Let {Yn, n > 1} be i.i.d. random variables with E(Y1) = 0, E(Y) =








1, and let {a,, n > 1} be a sequence of nonzero constants. If the series


E a < 0, (3.2.3)
n=l

then setting

o00
t2= = >
j=n

for every a > 0 and positive integer r, the tail series WLLN

sup IEi= a, Y
j>n P
------;a --+ 0
tn (log, t-2)

obtains.

Remark. The condition (3.2.3) is necessary for {Tn, n > 1} to be a well-defined

sequence of random variables where T, = Ej=, aj Yj, n > 1 (for clarification see

the discussion in Section 4.3 of Chapter 4).

Proof of Corollary 6. Let a > 0 and let r be positive integer. Set


bn = t, (logr t2), n > 1.


Since (3.2.3) ensures that t 1 0,

1 t ty 1
EE(ayj2) = t2 = --
b. t2 (log, t2)0 =) (log, t-2) 0

and so (3.2.1) holds with p = 2 where Xn = a, Y, n > 1. The corollary then

follows directly from Theorem 3. O

As a special case (r = 2 and a = 1) of Corollary 6, we obtain the following

corollary which will then be compared with a tail series LIL.








Corollary 7. Under the hypotheses to Corollary 6, the tail series WLLN
sup IEj ar I
t(o 2) --p 0 (3.2.4)
tn (log2 t;2)2
obtains.

The hypotheses of Corollary 7 (or Corollary 6) are weaker than those of some

results of Rosalsky [41, Theorems 2 and 3] which provided conditions for the tail

series LIL

lim sup j=n aJ Y = a.s. (3.2.5)
-o t (log2 tn2)
to obtain. Observe that the norming constants in (3.2.4) and (3.2.5) are the same.

The following two examples exhibit a sequence of norming constants {b,, n > 1}

for which a tail series WLLN holds, but a tail series SLLN does not. In the first

example, the harmonic series with a random choice of signs, which was considered

in Example 3 of Chapter 2, will be reconsidered.

Example 4. Let {X,, n > 1} be independent random variables such that


P IX-= =P =- 11 1 n>l.

Let 0 < a < 1. Then for arbitrary p E (0, 2]

E(IXIP) = n-P, n> 1.

Let r be a positive integer, and set

b, = n-I(log n)9, n > 1.

Then

E(IX.nl) n-P __ _
Sn f (log,n) (log








implying

lE( JP = n- (log, n)- = oo.
n=1 n=l

Hence, since p E (0, 2] is arbitrary, the hypotheses of Proposition 6 are not met.

Indeed, it will be seen below that for p = 2, r = 2, a = 1, {X,, n > 1} obeys

the tail series WLLN with norming constants {b,, n 2 1} but does not obey the

tail series SLLN with those norming constants (since it obeys the tail series LIL

with those constants).

Next, choose p = 2. Then

00 0) I 1

j=n j=n n

Thus for r > 1 and a E (0, 1],

1 00 n-1 1
E E(X?) o(l)
14 = n-l(log, n)O (log, n)o

ensuring (3.2.1). By applying Theorem 3, the tail series WLLN

sup ITj,
.>n P
-(log 0, (3.2.6)
n (log, n)2

obtains. Choosing r = 2 and a = 1, it follows in particular that

sup |1Tj
n-_l I -- 0. (3.2.7)
n 2(log2 n)

(Note that {X,, n > 1} are weighted i.i.d. random variables, i.e., X, = an Y,, n >

1, where {Yn, n > 1} are i.i.d. random variables with

1
P{Y,=1}=P{Yn=-1}=-, n>1
2 -








and an = n-1, n > 1. Thus, by applying Corollary 6 and Corollary 7, we can also

arrive at the same conclusions (3.2.6) and (3.2.7), respectively.) But, recalling the

tail series LIL (2.4.14) of Example 3 of Chapter 2, it is clear that the tail series

SLLN

T.
S- 0 a.s.
n-2 (log2 n)
fails.

Next, an example constructed by Rosalsky [41, Example 1] of weighted i.i.d.

random variables will be discussed in the context of the tail series WLLN and

SLLN.

Example 5. Let {Yn, n > 1} be i.i.d. random variables with E(Yi) = 0, E(Y,2) =

1. For each n > 1, let

2 (log0 n)" (log n)"
nexp {(log, n) (logn) (log3n)}, < <00

and assume that if u > 1, then


E(Y12 (log2 IY1 )') < oo for some q > u -1.

Then the condition (3.2.3) obtains (see Rosalsky [41, Example 1]). Then, for r >

1 and a > 0, the tail series WLLN
sup IEcij ai Yi

t (log, t2) (3.2.8)
obtains by Corollary 6. By choosing r = 2 and a = 1 (or by Corollary 7), it follows

in particular that

sup IEj" a l II
ji> P
j2 ----j 0 .
t (log t-2)2








But by Rosalsky [41, Example 1], the tail series LIL


limsup m= aJ = V a.s.
t (log, t2))

obtains. Thus the tail series SLLN

E' aj Y
1-+ 0 a.s.
tn (log2 t2)n

fails.

Remark. It may be noted that the hypotheses to the tail series LILs of Rosalsky

[41, Theorem 2 and 3] ensure immediately via the Chebyshev inequality that

T, P
-" 0.
t, (logz2 2)

Indeed, the hypotheses to these theorems of Rosalsky [41] always entail the stronger

conclusion
sup JTjl
j> P 1
-------+ 0.
t, (log, t;2)
This observation was already made after Corollary 7 concerning the tail series LIL

of Rosalsky [41, Theorems 2 and 3]. Apropos of Rosalsky [41, Theorem 1], the

observation follows immediately from our Theorem 3 by taking p = 2 and b, =

tn (log 0 2) n 1.













CHAPTER 4
TAIL SERIES STRONG LAWS OF LARGE NUMBERS II


4.1 Introduction and Preliminaries


As was discussed at the end of Chapter 2, there is a gap between the conclusion

of our tail series SLLN (Theorem 2) and that of the tail series LIL of Rosalsky [41,

Theorem 2]. So, it is natural to seek a tail series SLLN whose conclusion is more

akin to that of the tail series LIL of Rosalsky [41, Theorem 2]. To this end, we will

establish in Theorem 4 below a tail series counterpart to the following SLLN for

partial sums by Teicher [47].

Proposition 7 (Teicher [47]). Let 1 < p < 2 and let Sn = Ej=i Xj, n > 1, where

{X., n > 1} are independent random variables with

E(X,) = 0, E(IX.I ) < en < oo, Bn = je oo, n 2 1
j=1
where {e,, n > 1} are positive constants. Assume that

Bn+1 = O(B,).

If for some a E [0, ) and some positive constants 6 and e


SP{IX,j > B. (log2 B.)1-'} < oo (4.1.1)
n=1
and
oo E(X2IB 1Io B1)- E ( [eB(lg, B)- n= (B(log, B)'-)2

55








then the SLLN
S.
-' 0 a.s. (4.1.3)
B. (log2 B,)1-a

obtains.

Remark. A standard Borel-Cantelli argument reveals that (4.1.1) is a necessary

condition for the conclusion (4.1.3) to obtain. Moreover, while the condition (4.1.2)

is technical in nature, it is not at all ad hoc in that it is of the spirit of conditions

employed by Chow, Teicher, Wei, and Yu [15], Egorov [19, 20], Heyde [22], Klesov

[31], Petrov [37] (see Heyde [22] and the inequality (I) of Loeve [34, p. 209] for clar-

ification), Petrov [40, p. 303], Sakhanenko [42], Teicher [45, 46], Tomkins [48], and

Wittmann [50] to prove LILs (or SLLNs) for partial sums of independent random

variables. Moreover, the above authors also employed a condition in the same spirit

as (4.1.1).

It will be seen after the statement of Theorem 4 that this theorem will yield a

sharper result than that of the tail series SLLN of Theorem 2 of Chapter 2 when

the hypotheses of Theorem 4 are satisfied. Furthermore, as special cases of the tail

series SLLNs of this chapter, we will investigate the tail series SLLN problem for

weighted sums of i.i.d. random variables. In the weighted i.i.d. case, it will also be

seen after the statement of Theorem 5 that this tail series SLLN will narrow the

gap between the conclusion of the tail series SLLN and that of the tail series LIL

of Rosalsky [41, Theorem 2].








4.2 Tail Series SLLNs


For independent random variables {X,, n > 1} we obtain tail series SLLNs

below, which are counterparts to the SLLNs for partial sums of Teicher [47]. The

main result of this chapter, Theorem 4, may now be stated. As in previous chapters,

{T., n > 1} denotes throughout the tail series Tn = EC' Xj, n > 1, corresponding

to random variables {Xn, n > 1}. It will be shown in the proof of Theorem 4 that

the hypotheses guarantee that {T, n > 1} is a well-defined sequence of random

variables. But, the proof of Theorem 4 will be deferred until after the proof of the

ensuing Lemma 10.

Theorem 4. Let 1 < p < 2 and let {Xn, n > 1} be independent random variables

with

E(Xn) = 0, E(jXnj) < en, n > 1

where {en, n > 1} are positive constants with YE~ en < oo. Assume that

AP = O(AP+1) (4.2.1)

where At = E ej, n > 1. If for some a E(-oo, 0 )

X P{IX. > 6 A. (log, AP)1-} < oo for some 6 > 0 (4.2.2)
n=l
and for all e > 0

oo E (X eA(10g, A')-" E X ---< oo, (4.2.3)
=1 (A. (log2 An"P)1-a)
then the tail series SLLN

0 a.s. (4.2.4)
A. (log2 An-P)1-a








obtains.

Remarks. (i) Note that (4.2.1) ensures that A, An+1 > 7 for some y E (0,1).

(ii) Note that if {X,, n > 1} satisfies the hypotheses to Theorem 4, then the

hypotheses to Theorem 2 of Chapter 2 are also satisfied with


gn (x)- =xl, 1 < p < 2, n > 1

and so for any O(x) 6

T.
n- 1-* 0 a.s.

where
00oo
A = E(IXjlP), n>1.
j=n
As for as notation is concerned, note that if {X,, n > 1} obeys the hypotheses to

Theorem 4 with en = E(IXnP), n > 1, then the sequence {An, n > 1} of Theorem

4 is in fact the sequence An, n _> 1 However, in this case,


(log2 AP)1-a = ((Al)))

whence Theorem 4 yields a sharper conclusion than does Theorem 2. Of course,

(a) in general, the hypotheses of Theorem 2 may be satisfied, but not those of

Theorem 4,

and

(b) Theorem 2 involves a class of norming sequences which is structurally

different from that of Theorem 4.

As will become apparent, Theorem 4 owes much to the work of Teicher [47]. The

proof of Theorem 4 utilizes the following two lemmas. In Lemma 9, there are no as-





59

sumptions concerning the integrability of the random variables exp{t S}, exp{t S,},

n > 1, in (4.2.5). Moreover, Lemma 9 cannot be proved by invoking the continuity

theorem for moment generating functions unless S,, n > 1 and S are all defined on

a common interval of the t-axis containing 0 as an interior point.

Lemma 9. Let Sn, = E=Xj, n > 1, where {X,, n > 1} are independent ran-

dom variables with

00
E(X,) = 0, n > 1, and f E(XJ) < oo.
n=l

Then there exists a random variable S with E(S) = 0, Var(S) = E= E(X2) and

S, -- S a.s. and such that


im E(exp{t S.}) = E(exp{t S}), -oo < < oo. (4.2.5)




Proof. The existence of a random variable S with E(S) = 0, Var(S) = E'=i E(X2)

and S, -- S a.s. follows directly from the Khintchine-Kolmogorov convergence the-

orem.

Next, for all n > 1, Jensen's inequality ensures that


E(exp{t T.+i}) > exp{t E(T,,+)} = e = 1


and so


E(exp{tS)) = E(exp{tTn+}exp{tSn})

= E(exp{t Tn+1}) E(exp{t s,}) (by independence)

> E(exp{tSn}).








Thus,


limsupE(exp{t S.}) < E(exp{t S}). (4.2.6)


Moreover,


E(exp{tS})) = E(limexp{tS})

< liminfE(exp{t S}) (by Fatou's lemma)


which when combined with (4.2.6) yields the conclusion (4.2.5). 0

Lemma 10. Let {Xn, n > 1} be independent random variables with jX.nI

M,, n > 1, where {M., n > 1} is a bounded sequence of positive constants and

suppose that


E(X,) = 0, n > 1.


(i) If the series

E < 00, (4.2.7)
n=l
where aon = E(Xn), n > 1, then setting
00oo
t2 = a oJ a 2 1,
j=n

the inequalities


(exp T < ex2 1 + tC)}, n> 1

obtain for all t E (0, C1-] where

1
C, = -sup M, n > 1.
in j>n







(ii) In addition to the assumptions in part (i), let {z,, n > 1} be a numerical

sequence satisfying

0 < Cn, 5 u, n> 1 (4.2.8)

for some constant u < oo. Then the inequalities


P supTj > A t < exp {-zx' vA (1 + n> 1
I I 2 n 2

obtain for all A > 0 and all v E (0, u-1].

Remark. Observe that our A in part (ii) may vary with n, i.e., the above A can

be replaced by A,.

This lemma is a tail series analogue of the exponential bounds lemma of Teicher

[47, Lemma 1]. The proof of Lemma 10 employs the function 2-1 (1 + 2-'x) playing

a similar role as the function g (z) = x-2 (e" 1 z) of Lemma 1 of Teicher [47].

Proof of Lemma 10. (i) The argument is contained in the proof of Theorem 2 of

Chow and Teicher [13].

(ii) In order to prove part (ii) of the lemma, we will employ the argument in

the proof of Proposition 5 of Chapter 3. As in the proof of Lemma 7 of Chapter 3,

note that the hypotheses ensure that, for a given n > 1, {Sn,M, .,,M, M > n} is

a martingale where

M
Sn,M = E Xj, F ,M = a(Xn, ..., Xu), M > n > 1
j=n

and so for t > 0, {exp{(i Sn,M}, Fn,M, M > n} is a submartingale (see, e.g., Chow

and Teicher [14], p. 232) since the function y(s) = exp{t s} is convex. Then for








N > > 1, v E (0, u-'], and t = v ,, we have


P max Tj > Axz. t}
n:5j:N )


f M
= P max lim X, >AXzt.
n ( M
= P lim max Xi > A Xn tn}
Mf-oo n

I lim max -.xo>Antn
M-+oon
= E liminflr
IM- Ao max I -'.Xi>X-.xtn
Ln .

M-oO n<Awnt,.
t Ib__


(by Fatou's lemma)


=liminf P
M--oo


M
max Z X, >
n S=i


< liminfP max Xi > A z, t,
S< M-*oo

= liminf P max exp{- ,M } > exp{
M-oo njS liminf E(exp{( S.,M})
< m-.fo exp{t- z}
M-oo exp{tAaXn}


Xztn}} (t>0)


(by Doob's submartingale maximal inequality [18, p. 314])
E(exp({- Tn
=Eexp{It A1X}) (by Lemma 9)
exp {tLA z, )
< exp{-tAXn+ (l+ )}

(by part (i): note t E (0, C-1])

= exp -vAx + 1v (+ 2 v(, )


5 exp{ x vA


A Xn tn }


T(1 2)


} (by(4.2.8)).







Letting N -+ oo yields

P sup Tj > A z tn} = lim P max Tj > A x, t,}
itj>I J N-Koo f n < exp{-z( V2A- (1 + ))}

thereby proving the lemma. O

Proof of Theorem 4. Observe at the outset that the tail series {T, n > 1} is well

defined by taking g, (z) = IzxI, n > 1, in Theorem 2 of Chapter 2. (Alternatively,

with the above choice of {g, (z), n > 1}, Loeve's [34, p. 252] generalization of the

Khintchine-Kolmogorov convergence theorem ensures that {T, n > 1} is a well-

defined sequence of random variables.)

Let e > 0 be arbitrary and let 0 < a < 1. For each n > 1 set
p

Un = Xn I[Ix.<5eA.(log2 AnP)-.]

V, = Xn I[Ixl>6SAn(log, Ap)1-a]

W, = X, I[ An(log2 AnP)-~
Then X, = Un + V, + W,, n > 1. Now, for each j > n > 1,

E(I Vj) < E (ixi I[6Ai(log, AT P)I-
+ E(jXj| I[ix|>A.(log2 A;P)-a])

< A, (log2 A;P)-a P{IXj > bAj(log, A-)-)1}

+ A- (log2 A-;')(P-1) E(IXj [ixl>An(1og A-

and so

I|E()I| < A, (log2,AP)-" P{IXj, > 6Aj(log, A,-)1-}
j=n j=n








+ A-P (log2 A-P)(P-1) : E X I >An(lo A -a

= o(A- (log, AP)1-a) (since ap < 1), (4.2.9)

using (4.2.2) and the fact that

E X(l' I[ixijl>A,(10g2AP)-- <- A'.
j=n

Note that (4.2.2) ensures via the Borel-Cantelli lemma that a.s. Vn is eventually 0

and consequently so is t=, 1V. Thus

=n --j 0 a.s.
An (log2 An,)1-"

implying via (4.2.9) that

S{ I ( -- 0 a.s. (4.2.10)
An (log2 AnP)1-

In view of (4.2.3) and Khintchine-Kolmogorov convergence theorem


: Aj (logA- E ) converges a.s.
j=1 AJ (log2 Aj )1-

Then by applying Lemma 1 of Chapter 2 to this we obtain

E =~ { Wj E(Wj)}
S- W -+ 0 a.s. (4.2.11)
An (log2 An')1-'

Now, observe that E(X,) = E(Un) + E(V,) + E(Wn) = 0. Then, in view of

(4.2.10) and (4.2.11), in order to show that (4.2.4) holds, it suffices to show (since

e is arbitrary) that Rn = E J,, {Uj E(Uj)}, n > 1, is a well-defined sequence of

random variables satisfying

sIRn 6e
lim sup n A)-< 6- a.s. (4.2.12)
n--oo A (log, Ann)1-0 72








where 0 < 7 < 1 is as in Remark (i) after the statement of the theorem.

To this end, firstly observe for each n > 1 that


E(IU. E(Un)lp)


5 E((IU.1 + IE(UD)I)p)

< 2'{E(IUp'+ IE(Un)IP)}

= 2P{E(IUnP) + IE(U.)IP}

< 2P{E(JUIP) + E(IU|,')} (by Jensen's inequality)

= 2p+ E(IU.IP)


and so


f E(|U, E(Un)I )< 2p+I E(IUn) < 2p+1 E en < oo.
n=l n=l n=l
Thus, by taking g, (x) xIzl, n > 1, via Theorem 2 of Chapter 2, {In, n > 1} is

a well-defined sequence of random variables.

Next, recalling that An+1 > 7 where 0 < 7 < 1, let

nk = inf {n >1 : A < 7k, k >1.

Then, for all k such that nk > 2, since An,,k > 7k,

Ank 7 Ank-1 > 7 k+l AnA,.

Hence {nk, k > 1} is a strictly increasing sequence of integers. Moreover, for all

k > 2 such that nk > 2, since

An. > 7k+1 and A,_, < 7k-1,


it follows that


Ank 7k+1
An_, __ > 2


(4.2.13)







For each n > 1

P R> 6 A, (log2 AP)'- i.o.(n)
fI 7 n 1
SP n max R > An, (log2 A,)l i.o. (k)
I 6k21 SP sup R > A, (log, Ap-) i.o. (k)

P A (l A k)
< P,{ sup R > 6e A-,, (log, A:) l- i.o. (k)

(by (4.2.13) and the fact that A-P T as k T)

= Psup. > 6 e A, (log2 A~)1 i.o.(k)}. (4.2.14)

Now, for each n > 1, let r = E(Ri). Then

2r < E(XJ I[IXly ,A(1 Al )_-
j=n
= ~ E(x'lplXjI2-P I[Ixj<,Ae,(log, A'P)-a]
< 2-p A2-p oo
(log A)a(-p) =n
E2-P A, 2
= (lo A (4.2.15)
(log, AP)a(2-P)'

For each n > 1, note that, since Un 5 e An (log2 A;P)- ,

IUn E(U,)I < 2e A, (log, A-P)-' = Mn, n 2 1.

Then, for each n > 1, setting

1= M 2 e A (log2 A;~)-
Cn = upMj = -- sup=M
rn j>n rn rn
66e A (log2 Ap)1-2P
An r- n---- ---

rn (log1 A-P)'
z, A,








it follows that


CnhZn = 2e

Ank X n rnh = 6 e Ank (log, A-')1-

A n = 6 (log2 A-P) --+oo

2 rk (log, A;'-) 2
Xn = A 0 < 62-p (log2 Ag )"P = o(log2 AP)
Wht


(4.2.16)

(4.2.17)

(4.2.18)

(4.2.19)


by (4.2.15) and the fact that ap < 1.

Now by (4.2.14) and (4.2.17) we obtain

P Rn > A. (log2 A;-)1-" i.o.(n) < P sup Rn > A,,r, i.o. (k).
S(4.2.20)
(4.2.20)


But, for all k > 2 such that


P{ sup Rn > Anskrn,k
n>n* )I


nk > 2, letting u = 2e and v E (0, u-'], we have

< exp -X v (vA. -(1 + e v)

( by part (ii) of Lemma 10 recalling (4.2.16))

= exp-3 (log2 A;:) + K x (K = )

( by taking v = and by (4.2.18))

< exp{-2 (log2 A-))) ( by (4.2.19))

= (log A;:)-

< (-p klogI 7)-2 ( since A-P > 7-Pk)


and so for some constant C E (0, oo)

{ (00 lo1 00 1
P sup R, > A; Xnr < C + 2 E 1- < oo.
k=1l n"J p' (nlog 7)2 = k=








Hence, by Borel-Cantelli lemma,


P sup R. > An,,r,,r, i.o.(k)} =0

implying via (4.2.20) that


P Rn > A. (log2 Al P)1-' i.o.(n) = 0.

Hence
R_ 6e
lim sup ,< a.s. (4.2.21)
noo A (log, A')1-a -72

Since {-(Un E(U.)), n > 1} have the same bounds and variances as those of

{(U. E(Un)), n > 1}, (4.2.21) likewise obtains with -Rn replacing Rn thereby

proving (4.2.12) and the theorem. O

By taking p = 2 in Theorem 4, we obtain the following two corollaries which are

partial analogues of Teicher's [47] corollaries of Proposition 7 above.

Corollary 8. Let {Xn, n > 1} be independent random variables with

oo
E(X,) = 0, E(X ) = ao, n> 1, and t = j = o(l).
j=n
Assume that

tn = O(t+1). (4.2.22)

If for some a E (-oo, )

00
SP{IX.n > 6tn(log2 tV2)1- < 0o for some 6 > 0 (4.2.23)
n=l

and for all e > 0

oo E X(XI[ t(log2,;2)-_ E (lo-g < oo, (4.2.24)
n= tn (log 2)1-)2








then the tail series SLLN

T,
-+ 0 a.s. (4.2.25)
tn (log, tf2) -n

obtains.

Remark. Observe that this corollary precludes a = In fact, the conditions

(4.2.23) and (4.2.24) when a = comprise two of the three conditions for the tail

series LIL of Rosalsky [41, Theorem 2].

Corollary 9. Let {X,, n > 1} be independent random variables with


E(X,) = 0, E(X.) = ao2, n > 1, and t = 2 oJ = o (1)
j=n

If for some a* E (0, |] and M E (0, oo),


IX.I < M t,(log2 t;2)-' a.s., n > 1, (4.2.26)

then the tail series SLLN (4.2.25) prevails for all a < a*.

Proof. For a* E (0, 1], observe at the outset that


t2 t2
M2 t2 (log2 t-2)-2a*

M2
= 1-
(log, t2)2"*
1 (since 1t = o(1))

and hence (4.2.22) holds. Note that (4.2.26) ensures that the conditions (4.2.23)

and (4.2.24) hold for any a < a*. The corollary then follows from Corollary 8. O

The two conditions (4.2.2) and (4.2.3) of Theorem 4 will now be combined into

a single one in the next two Corollaries 10 and 11 which are comparable with the







tail series LILs of Rosalsky [41, Corollaries 1 and 3]. That is, a condition which

ensures that the conditions (4.2.2) and (4.2.3) are simultaneously satisfied will be

employed in each of the following two corollaries.

Corollary 10. Let 1 < p < 2 and let {Xn, n > 1} be independent random vari-

ables with

E(X.) = 0, E(IXj.I) < en, n > 1

where {en, n > 1} are positive constants with EC=, e, < oo. Assume that (4.2.1)

holds where AP = Ej, ej, n > 1. Let -oo < a < 1 and 0 < P < 1. If for all e > 0

oo E( Xg| I[IX.I>A.(log, A4-)-]
S(A (lo2 AP)-) 2 < 0o, (4.2.27)
n=l (An (log1 An-)1-_)
then the tail series SLLN (4.2.4) obtains.

Remarks. (i) Observe that a smaller a gives us a weaker assumption (4.2.27) as

well as a weaker conclusion (4.2.4).

(ii) Also observe that for 3 = 0, the condition (4.2.27) reduces to
oo
SP{IX, > e An (log, AP)-*} < oo for all e > 0
n=l
and for / = 1, it becomes
Eoo E(XI[j.>Ana(0g;A-)-
2 < oo.
n=1 (A. (log2 AP)'1-0)2
Proof of Corollary 10. Note that for all large n

E(IX.12 I[IXl>A.(log2 A)--]) E (jXn20 I[|X.l>A.(log2 An)1-])
(A, (log, A1')1-') 2 (A (log, A(1')1-.)2
> E(I[IxI>An(log, A-'-)-*]

SP{Xn, > An(log2 AP)l-'"}.







Then for some constant C E (0, oo),

EP{X,I > A. (log, AP)'-'}
n=1
Co E (lX 12P II[Ix=>An(log A)'_)-.])
n=1 (A, (log2 An')1-)2
< oo (by (4.2.27))

implying the condition (4.2.2) with 6 = 1.
Next, note that for arbitrary e > 0 and all n > 1

E (IX, 1" I[I|,.>eA,.(log, A;p)-])
(An (log2 A'P)1-")26


(A, (log, A-_)
E (iX, 2-2(1'-) [eAn(1log0 A;P)-
(An (log2 An')I-,)2-2(I-P)
SE(X2 I[.A(log AP)-a<|IXn
(A, (log, Ap)1-,)2
Then

SE(X 'A(10og Ap)-O n=1 (A. (log2 AnP)1-)2
0 E (lXn,2f I[|Xnj>.An(,log, A'P)-])
n=1 (A (loga An")1-)2-
< oo (by (4.2.27))

and so the condition (4.2.3) also holds with 6 = 1. The corollary then follows
directly from Theorem 4. 0







Corollary 11. Let 1 < p < 2 and let {Xn, n > 1} be independent random vari-

ables with

E(X,) = 0, E(IX, I) < en, n > 1

where {en, n > 1} are positive constants with E,=i en < oo. Assume that (4.2.1)

holds where A = ECjn ej, n > Let -oo < a < 1, and 7 > 1. If for all e > 0

Y (log1 A-P)-1^P('-"
E (IX I''[IX.I>eAn(log, A;P)-] = ( g e. (4.2.28)

then the tail series SLLN (4.2.4) obtains.

Proof. Note that for arbitrary e > 0 and all large n, (4.2.28) ensures that for

some constant C1 E (0, oo)

E(IXnlP I[IXI>An(log2 A;P)-]) ______
(An (log2 AP)1-a)P AP logici AnP) (log2 A;P) (log3 A )7"
Then for some constant C2 E (0, oo)

E E(Xn lP I[|X.|>'An(og, A,)-_)
n=1 (An (logz AnP) ')-
< c+C e
n=1 An (log, AZ') (log, An") (log3 An-')
< oo (by Rosalsky [41, Lemma 5])

and so the condition (4.2.27) holds with P = E. The corollary then follows from

Corollary 10. O


4.3 The Weighted I.I.D. Case

For i.i.d. random variables {Yn, n > 1} with E(Yi) = 0, E(Y,2) = 1, and for

nonzero constants {an, n > 1}, {an Y, n > 1} is a sequence of weighted i.i.d.








random variables. Then there exists a random variable S with EC= a- Yj -- S a.s.

iff E,= a' < oo. (Sufficiency follows directly from the Khintchine-Kolmogorov

convergence theorem whereas necessity results from the work of Kac and Steinhaus

[28] or Marcinkiewicz and Zygmund [35] or Abbott and Chow [1].) In such a case,

E(S) = 0, E(S') = E'= a .

Corollaries 8 and 10 reduce to Corollaries 12 and 13 below, respectively, in the

weighted i.i.d. case.

Corollary 12. Let {Y,, n > 1} be i.i.d. random variables with E(Yi) = 0, E(Y2) =

1, and let {an, n > 1} be nonzero constants satisfying t2 = EC a = o(1) and

(4.2.22). If for some a E (-oo, ,)


SP{lYi > 6 lajl- t (log, 2t2)1-" < oo for some 6 > 0 (4.3.1)
n=l

and for all e > 0

oo anE aInl-'tn(log, tn2 )-@11<6-nn(l10g, t;2)1--])
E --E --- 2 < oo, (4.3.2)
n=1 t. (log, t;2)1-

then the tail series SLLN

E;=0 aj
S- 0 a.s. (4.3.3)
t, (log, t 2)1-a

obtains.

Proof. Since the conditions (4.3.1), (4.3.2), and (4.3.3) are simply transcriptions

of (4.2.23), (4.2.24), and (4.2.25), respectively the corollary follows immediately

from Corollary 8. O

Corollary 13. Let { Y, n > 1} be i.i.d. random variables with E(YI) = 0, E(Y 2) =

1, and let {a,, n > 1} be nonzero constants satisfying t2 = E a = o(1) and








(4.2.22). Let -oo < a < 1 and 0 < P < 1. If for all e > 0

aoo la2 E (lYI1 I[y l .lai-t(log2 t;2])
2 < oo, (4.3.4)
"=' (t. (log t-2)1-")

then the tail series SLLN (4.3.3) obtains.

Proof. Since the condition (4.3.4) is a simple transcription of (4.2.27) with p = 2,

the corollary follows directly from Corollary 10. 0

The main result of this section, Theorem 5, which is an analogue of the tail

series LIL of Rosalsky [41, Theorem 2], may now be stated.

Theorem 5. Let {Y,, n > 1} be i.i.d. random variables with E(Yi) = 0, E(Y2) =

1, and let {an, n > 1} be nonzero constants satisfying t2 = E ,a = o(1) and

(4.2.22). If
2
nan = ((log2;2)) for some -oo < < oo, (4.3.5)

then the tail series SLLN

-+ 0 a.s.
t, (log t~2)1-"

obtains for every a E (-oo, 1) provided in the case 7 > 2(1 a) that


E(Y12 (log, IYl)-2(1-a)) < oo. (4.3.6)



Remark. Actually, under the assumption (4.3.5) where r < 1, the result follows

directly from the tail series LIL of Rosalsky [41, Theorem 2]. In the case 1 <

r < 2(1 a), the additional assumption is not needed in Theorem 5, although an

alternative additional assumption in the same spirit as (4.3.6) is required for the








tail series LIL of Rosalsky when (4.3.5) holds with r > 1. And for r > 2(1 a), we

assumed the moment condition (4.3.6) which is weaker than the additional moment

condition in the tail series LIL of Rosalsky since 7 2(1 a) < r 1.

Proof of Theorem 5. Without loss of generality, it may be assumed that r > 0

and a > 0. Note that (4.2.22) is tantamount to tn2 t2_1 < M1 for some constant

Mi E (0, oo) and all n > 2. Then for all n > 1,


t-2 = t-2 < tMn-1 = O(Mn)
j=2 I

implying

log2 t2 < (1 + o(1)) log, n. (4.3.7)

Moreover, for all large j and for some constant M2 6 (0, oo), observe that

j-1 + -1 j-1
= 14

(log2 t- '
S1+ Ms2 --( (by (4.3.5))
j-1

< 1+2M2(log 1)), (by (4.3.7))
j-1



and so for all large no and n > no

no = l til
t2 j=no+l J31

< II (1+2M(j 1)-1 (log, (i 1)))
n-1
= I exp 2M2j-1 (log1 j)}

-exp J n( 2M oj- J)







n-1
exp 2 M2 (logn)r j-

5 exp({2M2 (log1n)T7+1.

Thus

log 2 = O(log2 n). (4.3.8)

In view of Corollary 13, it is suffices to show that the condition (4.3.4) is satisfied

for some / E [0, 1] and all e > 0. Recalling (4.3.5), let K E (0, oo) satisfy

2 K(log, t2 ) (4.3.9)

for all large n. For each n > 1, let

e2 n
q K(logt2) (4.3.10)
K(log1 09 )'+2

where e > 0 is fixed but arbitrary. Then, by (4.3.9)

e2 tP
S< (log t (4.3.11)
a2 (log, t2)2a

and also by (4.3.8) and (4.3.10)


log t0 2 = O(log2 qn). (4.3.12)

Next, it will be demonstrated that {q,, n > 1} is eventually nondecreasing with

q, -- oo. To this end, note that

n ( ,2 t- ) n a'-1
t-l (logl t-l2) (log-,2) 2 (log, t ,) (log, t-2)
__ (1) (n 1) a___
o2 _) (by (4.2.22))
-1 (log t n-0) log2,n-)
= o(l) (4.3.13)





77

by the assumption (4.3.5). Let O(x) be the extension of {t2, n > 1} defined by

linear interpolation between integers, i.e., for all n > 2

n(x) = t nln + (t-t1) (x n + 1), x E[n- 1, n).

Then, via the mean value theorem, for all large n there exists a number z, in

(n 1, n) such that

2 2 2n e2(n-1)
-q qn-1 K(log, t2),+2a K (log2 t2i),+2

(e 1 (7 + 2a) (t-2 t2x) }
K (log, 2 (X,))+2 (X ) logici ,(x,,))(log2 (X))++

= ~-(1 + o(1)) (log ))+2 (by (4.3.13))
K (log, (o2 (X,))
2K
>2 (log2 (z,))-( +2)

> 0.

Thus, since (4.3.10) and (4.3.8) guarantee q2 -+ oo, we have verified that {q,n, n 1}

is eventually nondecreasing with qn -- oo.

Let 0 < # < 1 and r* > 0, the exact choices of which will be made later. Then

a.| l1'2 E (I 2[ ,'1 Y>l>-n1t.(log2, t
n=1 (t (log, 2) )2
oo E (IY12 I['>1,>I ,lt"(log2 2)-])
n 0(1)(l 2og )2p(-)-OP (by (4.3.9))
n=1 n (log) t)
5 O(i) n E(Pg2z[ -)-) (by (4.3.11))
n=l xn (log10 t2 )
< 0(1) E '"mR E(IY1X1 iE5 zn=1 n (log, t2)P(2(1-a)-r)

= O(1) EE( 2 Il,, +) n(log2 t-2)fn(-2(1-a))
j=1 n=l








O(1) E j-- (log2 t'2) (r-2(1-a)) E(Yi12 I[q j=1
(by Rosalsky [41, Lemma 6])
SO(1)---logt -2t *(,-2(1--)) E(Y12(log2 1YID)* I[qi j=1 qj (log2 qj)*

= O(1) (lot 2)`(r-2(-+1-a))(l-P)(r+2a) E(Y12(log12 1Y11) I[qi, j=1 (log, qj)r*
(by (4.3.10))

< O(1) E (log2 t 2)'+2 E(Y12(log,2 1Y1,) I[j, j=1
(by (4.3.12)).

If r < 2(1 a), let r* = 0 and f = '(r + 2a) and then via (4.3.14) the series

of (4.3.4) is dominated by 0(1) E(Y12) < oo.

Alternatively, if r > 2 (1 a), let r* = r 2 (1 a) and f = 1. Then again via

(4.3.14), the series of (4.3.4) is dominated by 0(1) E(Y12(logz2 IYl)') < oo recalling

(4.3.6). The theorem follows then directly from Corollary 13. 0

To illustrate Theorem 5, we will revoke previous examples from Chapters 2 and

3.

Example 6. As was observed in Example 3 of Chapter 2, the harmonic series

with a random choice of signs yields the tail series LIL (2.4.14) thereby ensuring

the tail series SLLN

(log, n)+T 0 a.s. (e > 0)
(log2
obtains. Alternatively, the same conclusion follows directly from our tail series

SLLN (Theorem 5 with r = 0 and a = 1 e).

Example 7. The same argument as in Example 6 can be applied to Example 5








of Chapter 3, i.e., since


~ (log, n)" (log3 n)" and log2t log2 n
n

(see Rosalsky [41, Example 1] for verification), the condition (4.3.5) holds with

7 > u (or 7 = u if v < 0) and so from Theorem 5 the tail series SLLN


-- 0 a.s.
t,, (log2 tn2)1-a

obtains whenever -oo < a < 1 provided (4.3.6) holds if 7 > 2(1 a).

In the following example, we will see an application of the tail series SLLN to

the field of time series analysis.

Example 8. Let {St, t = 0, 1, 2, ...} be the moving average process of

infinite order given by
00
St= E a Xt-j (4.3.15)
i=0
where {Xt, t = 0, 1, 2, ...} are i.i.d. normal random variables with mean 0

and variance 1 and {ai, j > 0} is a square summable sequence of constants. As a

specific example, consider a long memory process, which is represented by (4.3.15)

with ao = 1 and

r(j +d) k 1 + d
a, = k. = ,j1 (4.3.16)
-(j +1)l'(d) o
where
tx-le-tdt, if X > 0
Jo
1
|d| < 2 and r(x) = oo, if x = 0

z-1 r(1 + x), if z < 0.








By Stirling's formula


r(z) ~ 2 e-,+1 (x 1)'-1 as z -- oo


applied to (4.3.16), we obtain (see e.g., Brockwell and Davis [11], p. 466) for d # 0

jd-1
aj ~ r(d) as -

Then
oo 01 j( 2d-1
t2= _a2 T n
S ( j))2 (d-) (1 2d) (r(d))2
j=n (r(d))a j=n
implying
n2 na2
log2 t~ log2 n, = O(t1+x), and = O(1).
-n

Thus the conditions (4.2.22) and (4.3.5) (with r = 0) hold and so for every integer

t and all a E (-oo, 1), the tail series SLLN

n2I-d 00
(log2 n)_ aj Xt-j -+ 0 a.s.

follows from Theorem 5. We have thus determined an order bound on the almost

sure rate in which E o aj Xt-j converges to St for every t. Observe that this rate is

independent of the time t. Of course, E=o aj Xt-j is structurally far simpler than

St.













CHAPTER 5
SOME FUTURE RESEARCH PROBLEMS


The current research work suggests a number of open problems or areas for

future research activity. These problems or areas are discussed in this chapter.

In Chapter 2, a function O(z) in a specific class of functions T defined by Klesov

[29 and 30] was employed for determining the norming constants for tail series

SLLNs for random variables. Actually, this class 9 is a tail series partial analogue

of the class ck defined by Petrov [38] as follows; a function f belongs to Ic if it

is a positive and nondecreasing function such that the series E = converges.

In the case of the SLLNs for partial sums, Egorov [21] defined a wider class of

functions F, as follows; a function f belongs to F, if (f(x))' E Tc for some e > 0. In

the same spirit as in Chapter 2, tail series SLLNs which might exist and correspond

to Egorov's [21] SLLNs for partial sums will possibly employ a function in a class

which is a tail series analogue of Fc. It would be particularly interesting to see

whether such tail series SLLNs subsumes the results of Chapter 2.

In the Theorem 4 of Chapter 4, we established the counterpart to the SLLN for

partial sums of Teicher [47]. But Theorem 4 is indeed an incomplete analogue of

the SLLN of Teicher [47] because we assumed

oo E(X I[An(og AP)-a 2< oo, (5.0.1)
.=1 (A (log, AAP)I-)

81








for all e > 0 rather than for merely some e > 0, as was the case in a partial sum

version of condition (5.0.3) which was used by Teicher [47] to prove a SLLN. The

reason for this is that our tail series exponential bound (part (ii) of Lemma 10 of

Chapter 4), which is employed to prove Theorem 4 of Chapter 4, was proved only

for all v E (0, u-1] rather than for all v E (0, oo). Thus, by establishing an extension

of this exponential bound lemma without the restriction on v (as is the case in an

exponential bound for partial sums), the assumption (5.0.3) for all e > 0 might be

able to be weakened to (5.0.3) for some e > 0. Conceivably, under no additional

conditions or under mild conditions, the convergence of the series in (5.0.3) for some

e > 0 guarantees convergence for all e > 0 but this would require some investigation.

Next, it will be a very interesting problem to establish tail series analogues

of Adler and Rosalsky's [3, 4] general SLLNs for weighted sums of stochasticallyy

dominated or i.i.d) random variables. Adler and Rosalsky [3] established general

SLLNs of the form

Jj=1 aj (Yj 0 ')
-0 a.s.

where {Y,, n > 1} are stochastically dominated by a random variable IYI and

{7, n > 1} are suitable conditional expectations or are all 0. In their follow-

up paper, Adler and Rosalsky [4] provided sets of necessary and (or) sufficient

conditions for {a, Y., n > 1} to obey the general SLLN of the form

CF=I aj Y,
-+ 0 a.s.

where {Y,, n > 1} is a sequence of i.i.d. mean 0 random variables and {a,, n > 1}

are nonzero constants.








Finally, tail series problems for almost surely convergent series of independent

random elements taking values in normed linear spaces is an area ripe for extensive

research activity. Beginning with the pioneering work of Mourier [36] (wherein an

analogue of the classical Kolmogorov SLLN was proved for sums of i.i.d. random

elements taking values in a real separable Banach space), an extensive literature

of investigation has appeared on the SLLN and WLLN problems for partial sums

of Banach space valued random elements. For some recent developments in this

general direction, see the articles Adler, Rosalsky, and Taylor [5, 6, 7] and some

of the references contained therein (specifically, see Beck [10], It6 and Nisio [26],

H0ffmann-Jorgensen and Pisier [25], Woyczyniski [51, 52], Kuelbs and Zinn [33], de

Acosta [2], and Wang and Bhaskara Rao [49]). The necessary background material

for reading the above papers on Banach space valued random elements may be found

in Taylor [44]. Tail series versions of some of the results in the literature cited above

would certainly be a worthwhile research accomplishment. Indeed, the very question

as to when a series of independent Banach space valued random elements converges

almost surely is one which requires more investigation. For some results in this

direction, see Hoffmann-J0rgensen [24], Jain [27], and Woyczyiski [51, p. 386-390

and p. 430-431].














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BIOGRAPHICAL SKETCH


The author was born on June 24, 1956, in Kimcheon, Republic of Korea. In

1979, he graduated from the Air Force Academy, Seoul, Republic of Korea. He

was awarded a Bachelor of Science degree in mathematics in 1982 and a Master of

Statistics degree in 1985, both from Seoul National University, Seoul, Republic of

Korea. He then served as a full-time instructor in the Department of Mathematics

of the Korean Air Force Academy until 1988. He has held the rank of Major in the

Korean Air Force since 1987. He has published a paper (joint with Jong Woo Jeon

and Suk Ki Han), "Some Distribution Free Tests for Exponential Distributions,"

Journal of the Korean Society for Quality Control 14 (1986), 39-46.

Since 1988, Mr. Nam has been working towards the Ph.D. in statistics from the

University of Florida. He has been a member of the American Statistical Association

since 1989. He is married and has two children.

After graduation, Mr. Nam will rejoin to the Faculty Board of the Korean Air

Force Academy as an Associate Professor of Mathematics as well as a Lieutenant

Colonel of the Korean Air Force. His research interests lie in the field of limit

theorems for sums of random variables.








I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.




Andrew Rosalsky, Chaian
Professor of Statistics


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.




Rocco Ballerini
Associate Professor of Statistics


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.




Malay Ghosh
Professor of Statistics


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.




Richard Scheaffer
Professor of Statistics








I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.




Murali Rao
Professor of Mathematics


This dissertation was submitted to the Graduate Faculty of the Department of
Statistics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.




December 1992
Dean, Graduate School















































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