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SOME LIMIT THEOREMS FOR WEIGHTED SUMS OF RANDOM VARIABLES BY ANDRE BRUCE ADLER DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA PARTIAL FULFILLMENT FOR THE DEGREE OF THE REQUIREMENTS OF DOCTOR OF PHILOSOPHY ACKNOWLEDGMENTS would like to thank Andrew Rosalsky dissertation advisor, not only for his overwhelming assistance but also for his friendship. Also would like to thank Dr. Malay Ghosh, Ronald Randles, and Dr. Murali Rao for serving on my committee. For serving on my like thank qualifying to thank Ms. Cindy examination Rocco Zimmerman and oral Ballerini. defense In addition, for her expert typing committees, would of this would like to dissertation. would also like to thank my parents for their support which helped me in reaching my goals. Finally, would like to thank Dawn Peters was always there when needed her. TABLE OF CONTENTS Page ACKNOWLEDGMENTS. ... .. ..... .. . . ...........................ii ABSTRACT. . . . . . ............ ..... .................iv CHAPTERS INTRODUCTION .......... ....... ............ ................. 1 GENERALIZED CENTRAL LIMIT THEOREMS........................6 2.1 Introduction.................... 2.2 Preliminary Lemmas.............. 2.3 Mainstream.................... *. a a *t t *. ......... a 2.4 A Properly Centered Central Limit Theorem. 2.5 Asymptotic Negligibility................. 2.6 An Asymptotic Representation for {B n > 2.7 Examples............................... . THREE ........6 ........8 S..... .13 . ....36 .......50 .......52 .. ....67 GENERALIZED STRONG LAWS OF LARGE NUMBERS.................74 3.1 Introduction...... ..................................74 3.2 Preliminary Lemmas..................................75 3.3 Generalized Strong Laws of Large Numbers for Weighted Sums of Stochastically Dominated Random Variables........ .. .......... ......... ....... 84 3.4 Generalized Strong Laws of Large Numbers for Weighted Sums of Mean Zero Random Variables........105 3.5 The Petersburg Game...............................119 3.6 Examples. .............................. ... .......138 FOUR A GENERALIZED WEAK LAW OF LARGE NUMBERS................. 151 4.1 Introduction..... ............................ .....151 4.2 Mainstream...... ............... .. ......... ...... .151 Some Interesting Examples...........................167 REFERENCES. . . . . . ........... . . . . ..180 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 SOME LIMIT THEOREMS FOR WEIGHTED SUMS OF RANDOM VARIABLES BY ANDRE BRUCE ADLER May, Chairman: 1987 Andrew Rosalsky Major Department: Statistics Asymptotic behavior of normed weighted sums of the form n ak(XkYk )/b is studied. Central limit theorems as well strong and weak laws of large numbers are obtained. Firstly, we establish a generalized central limit theorem + N(0,1 assuming the {X, are independent, identically distributed with EX = 0, = mw The truncated second moment assumed to be slowly varying at infinity. Then, via a transformation, we obtain a similar result where the condition EX is removed. The norming sequence {b is defined in terms of n, n n the common distribution, and asymptotic representation for conditions which elicit an explicit are found. . I I j n I t A n * Xn, n 1} are stochastically dominated by a random variable X, while in others the random variables identically distributed. Two there are independent, ;ms are proved showing that the assumption of independence in general, not always needed to obtain a strong law. More specifically, their hypotheses involve both the behavior of the tail of the (marginal) distributions of the and the growth behavior of the constants b special cases, both old and new results are obtained. Moreover, independent, identically distributed a strong law established under the assumption of regular variation of the tail P{IXl of common distribution function. The famous Petersburg paradox is also examined in more general terms. It is shown that P{lim nl* k=1 aX /b k k = c) = 0 for any sequence and finite nonzero constant c whenever the random variables are independent, identically distributed with EIXI = c and la n' P Finally, a generalized weak law of large numbers (W obtained. S0) Using this theorem we are able to show that the modified Petersburg game does indeed have a solution in the "classical" or "weak" sense, i.e., there exists a sequence {bn, *n 1} such that + c, where the requirements placed on are as before. CHAPTER ONE INTRODUCTION This dissertation will studied in probability theor explore three major modes of We will convergence investigate both the weak and strong limiting behavior of a normed sum of weighted random variables. Such sequences of normed sums are expressed in the form akXk y, ) While the sequences and {b ,n > 1} will always be numerical, the sequence will consist of random variables which may distributed. forms, or may not be independent The sequence ranging from conditional expect or even I} will identically take on many different stations to all zeros. Chapter Two will examine when converges in distribution to a standard normal random variable central limit theorem (CLT). This is commonly referred to A sequence of random variables, as a say Z , converges in distribution to a standard normal random variable if i v ,,  , n > case when the random variables Xn, n 1} are independent with finite second moment has been thoroughly investigated (see Theorem 2.1). The topic of interest in Chapter Two is whether this limiting behavior still prevails when the second moment is infinite. We will only consider the situation in which the random variables are independent, identically distributed (i.i.d.). first major result will establish a CLT for mean zero random variables. In contrast to the case when the second moment is finite where the norming constants are universal (up to a multiplicative constant), the norming constants } when the second moment is infinite are not universal but, rather, depend on the common distribution of the 1Xn n Then by studying the shifted sequence of random variables = X EX n we will obtain a CLT for random variables with arbitrary first moment and infinite second moment. Strong laws of large numbers (SLLN) are investigated in Chapter Three. The sequence in is said to obey the strong law of large numbers if ak(Xkk)/b  0 almost surely (a.s.), that is, ak(X ~kk P{lim n*w = o = 1. This chapter contains many generalizations of the classical n Kolmogorov SLLN ( E X,/n 4 a.s. whenever * VI'  are i.i.d. work. Stout (1974, Chapter 4) for an excellent survey of known results on the SLLN problem for weighted sums of Mathematicians have, over the centuries, tr random variables. ied to understand the Petersburg paradox. Given a game with Xk winnings at the kth stage, it was asked if the game could be made "fair" in the sense that there is some entrance fee, b bk k k such that X /b k a.s. This is relatively easy when are i.i.d . with but is a source of confusion when E = C. This problem, when the random variables are not integrable, is called the Petersburg paradox. Placing mild restrictions on the sequence {a n 1n we will prove that akXk P{lim = c} = 0, every sequence {bn, n every finite nonzero constant provided that the random variables n' n 1 } are i.i.d. with This result generalizes one of Chow and Robbins (1961 wherein ak S1. For a detailed discussion of this paradox, Feller (1968, 251253). Feller , however, does produce a sequence {b n n 1 such that X /b k1 for a specific sequence of i.i.d. random variables with EIX Thsrp i ls sl ?n t.hp n.i ti. h n fll it J T I' It II a general i zed law of" = a. see = 00  i ir i no gn i i7 1 i u I P{lim = 1. See Rosalsky some results of this type. In the final chapter we consider the weak law of large numbers (WLLN). The normed sum ( is said to obey the weak law of large numbers Yk)/b that = o( for all e > Clearly, a SLLN holds, then a WLLN also holds for the same sequences. Hence Chapter Three also establishes many weak laws. view a very Criterion see, general e.g., result Chow known and Teicher, as the Degenerate 978, . 32 Convergence , we acknowledge the fact that most wor k on the WLLN for independent already been done. Research in the central limit theorem and laws of large numbers date back to the 18th century. very first SLLN was proved Emile Bor el, while the WLLN dates back much earlier. WLLN for i .d. random variables with a finite second moment established Jacob Bernoulli and published posthumously nephew Nicholas a simple appli cation of Chebys hev' i nenua i tv was shown that when n> was I I I  . J. A,  . .rL CLT. For a detailed history of the development of these and related concepts, see Feller (1945) and Le Cam (1986). Also, for an excellent survey of known results, Some remarks about notation a see Petrov mre in order. symbol C will denote a generic constant necessarily the same one in each appearance. (1975). Throughout a) which Also, is not in order to avoid minor complications = logemax{e,x}, e it proves convenient to define where loget denotes the natural logarithm. Finally, for x > N log2x will be used to denote log log GENERALIZED CHAPTER CENTRAL LIMIT THEOREMS Introduction The question as to whether a sequence of random variables obeys a CLT has a long and rich history. Firstly, in the 700s , there DeMoivre and Laplace discovering a primitive version of the CLT for sequence 930s of Bernoulli , Lindeberg, random Lbvy variables. , and Feller hundred established years what later we now generally refer as the CLT. This famous result, which we will state as Theorem 2.1, is known as the LindebergFeller theorem. Theorem (Chow and Teicher 291). ,n > independent random variables with , Var(X satisfying 2 EX I( k for all E > Ek ) k where = o(s then k=1 d + N(O was , in now are , n > Proof. See Chow and Teicher p. 290293 Clearly, this However, theorem L&vy is not applicable Khint chine when 935), = for and Feller some 1935) have studi the CLT problem when the random variables ,n > are i with = 00 Their version of the CLT stated as follows. Theorem 2.2 (Chow and Teicher, p. 300). If { .d. random variables with = 03 then nd  N(0 some 2'P{I C P{l c EX2 c+w EX I = 0. Moreover, for all n > so that be chosen EX I , while as the supremum of be taken = nEXI( Proof. Chow and Teicher , p. 300 02). Theorem was first proved using the function in (4) while our extension will use the function in (5 inst ead. are and An variables, = 0, Var(X ) = 1, 1 nonzero constants with 4 m and a an easy application of Theorem , e.g., Chow and Teicher, 978, P. 3 02). Preliminary Lemmas To study the CLT problem when a = =, one must examine a special class of functions. A function is said to be slowly varying (at infinity) if for all g(sx)  g(x) as x Closely related to slowly varying functions are those that are regularly varying. function h is said to be regularly varying with exponent p if h(x) for some slowly varying function useful property of every positive slowly varying function is that (see Rosalsky, 1981) log g(x) (and hence = o(lo = o(X ), g(x) as x  m all a > The question at hand sequences of constants variables is what condition should we place on the and the i.i.d. random 1 } to ensure that ak(Xk k k see = o( = xPg(  N( rate the variance of the variables approaches infinity. This where the notion of slow variation will be applied. We define H, truncated second moment of X, H(x) = EX2I(IX Another function of interest is G defined by G(x) 2tP{ Xj t}dt, The functions G and H were used by Rosalsky (1981 to prove generalized law of the iterated logarithm for weighted sums with infinite variance. The first lemma will be used to establish a relationship between H and G. Lemma 2.1. For every random variable and positive constant p, tPIP{ xl t}dt = sPp{ XI s} + EXP I(Ixl for all Proof. Let F denote the distribution function of the random variable that Flxl(t) = P{ x For s > integration ,.. aa  tPIp t}dt tP( FixI(t = [tPp p tPdF xl(t)] Xl = 1s P Using this lemma, with = 2, we obtain = x2P{ should be noted that is nondecreasing, continuous, as x + o (see Rosalsky, 981). Other relationships between and H which will be used throughout this chapter are given in the next lemma. Lemma 2.2. (Rosalsky, 981). The following are equivalent: G is slowly H is slowly G(x) ~ H(x), varying, varying, x2P{ x}/G(x) as x + w, + H( = o( At this variable point we note is not in m. an interesting then fact. the following lem If the random ma it is in p all 0 provided H equivalently is slowly varying. Lemma 2.3. If H is slowly varying then E ap for all 0 Proof. Lemma we obtain whether or not X S^p This follows considering the two cases; o or E Let 0 2p. Then via (3 Lemma and the hypothesis that is slowly varying, there exists a number such that if t , H(t) and t t}/H(t) Thus, or P whenever Hence o tp1p > t}dt + pit t}dt ptp = tp since _ P tP+a2  p p+a2 o . D One thing to note is that slow variation of H ensures narti nular that I. 1 I  This allows us to estbl ish a 1 mma wh i h tP+a3dt + pit p+a2 I Lemma 2.4. any random variable constant xl I(x = cP{ t}dt. Proof. We apply Lemma twice, first with = and then both cases we let II > = 1. Therefore, t}dt = cP{  cP{ t}dt. Before we proceed with our main result of this chapter, conclude this section with one last lemma. Note that the hypothesis of Lemma ensures, via (3) that Lemma If P{ is regularly varying with exponent then xlI( x )op P+1 as c Proof. We apply the following result in Feller 971, 281): E 4 0 = C; Using this and Lemma 2.4 we obtain = cP{ t}dt = cP{ + (1+o( 1)cP p+1 Mainstream With prove these preliminaries our first major ace result counted for, , but first we are ready we need to state a version Theorem which establishes a CLT for triangular arrays. Theorem 2.3. Suppose that for each n the random variables are independent and satisfy = 0, E2 = EX ,n nk nf* k=1 CSn) n holds for all s > then  N(0 Proof. Se , e.g., Billingsley 31031 p+c "p+! . ,X in ( and (5) respectively. Let Q(x) = x2/G(x and since slowly varying , it is clear, via ( that Q(x) + 0 can also be shown that = (see Rosal Lemma sequence is defined Since we note that next sequence interest is { , n > which sequence of partial sums of the squares of the weights, Finally we define the positive constants n 2 = nq first main result follows. Theorem 2.4. be i .d. random variables with = = and 2 x PH( X H(x) = o(1) as x + 0. Let {a be constants such that max 1 *e, via  G( Then akXk  N(O where , n > is defined as in (7) akEXI( Proof. Via Lemma 2.2, we see that implies that is slowly varying and that  H(x + ( (since Note that since and hence G(qn G(qn) = Q(qn) B2 n 2 2 = q /G(q ) n (7)). Utilizing G(x) m and + = we obtain via (12) that s2/B Noting that Snqn n n snq n n max 1 1 max a,i 1  s = ) = o( and G is nondecreasing, it follows that Snqn n n n min Snq n n > G(q in max Using the fact that is slowly varying together with and (15) obtain Snqn n n  G(q /n min sq n n /n max Next we will show that a2H(B k Since  1. is nondecreasing, a2H(B k 1 >1 SB2 B n aH( B ak max snqn nfn /n max sq nfn Lemma 2.2) n  "( B2 n , n >  G( n B Similarly, a H(B k a H(B k 1 = nH 2 B n Snq nrn Sn q nfl 1 n n~ = 1+o( Combining these two results yields (17). Define (n) X k ak , k k' = Var(S' Ii ,n > Now via Lemmas I' I I K  +o(1 w ! n 1 2 2 ak (EXI( X 2 k B k=1 n 1 2 2 B k=1 n SB/a, ))2  n k = (E X)2 B n = o(1) (by (14)). Using this result and (17) we conclude that Y  B2 as follows: 2 2 1 Y /B =  Var( n n B2 akXk I( akXk k k k k  n 1 2 = a Var(XI( X B k=1 SB n/lak)) 1 2 2 ,  = 12 I akEX I(XI B k=1 n 1 u2 = 1 a H(B /l a ) B k=1 n Bn / ak I 1 2  2 a (EXI( X B k=1 n 1 2 B I aa (EXI( XI B k=1 n < B / ak a SBn/lak))2 ))2 1. Next, noting that X E I and 2a (nEX2 a (EX ) 2a (EXI( a (EXI lXi < B /lak))2 max 1 max 1 2 E a, (EXI(IXI k=1 < Bn/ ak))2 S2(EX )2 n xS and hence (n EX k max 1 can now establish that the normed sum of the truncated (n) a (Xk k k Ex k variables, , converges in distribution standard normal random variable. Let E > 0. Clearly and ( ensure that for all large akE(n a EX, k k E/2. Hence for all large n and (n) EXk k < 2 or  2 ak equivalently ) EY , EX n) and EX(n Ek Thus for all large and all EXn 'k S(Cn) ( xk EXn ) EYn / a n k k k(nX U{ X k EXn ) EXn) EXk "k Y n U ixn) EX~n) k cY n 21a k .. (n max S(n) k = o( 2 k / ak n k / a, n k l(n) {Xk Uxn) Thus, for all , and all large E(a Xn Eak akn) k k Eak X, k k E,) n < 2 2 B n 2E (n EXk k 2 (n) )2I( Xk (n EX k (since 2 n a2 (n k k (EXn EXk k C(n Ik ) Cf ) I ) . the previous observation) < 4 B2 n 2 (n) a E((Xr k k + (EXn) k l (n) IXk (using the elementary inequality +b2) (ab) _4   B2 n 2 a E(X k kc _4 +  82 n (EXn (EXk k(n) Xk This which second is o( sum is o(1 since , recalling it is dominated S 4 2 )2 _2 Sn(EIXl The first sum is equal EY n I , 1 and ( B2 B (for all large n since Therefore by Theorem ES' n Let 6 + N(0 Then akXk k k (n) akXk <_P{k=1 k=1 'jak = o(1) = o( Hence ES' n n B n Y n + ( B n ES'  N(O (19), and Slutsky's theorem). Finally, noting that akn) k k the conclusion follows. With conditions this result in hand is A it is natural It would seem to ask, natural under assume what e that , but that in addition to the other hypotheses of Theorem does seem ensure that ak k k + N(0 T C' 0. 4 a A 4e 44 n S' +S' ES' = o( T+ t.ni 1 r\Q ca l^/Mr '> * + ^ Tn nhn ftr) \f /^ / irt C1^^^ s r nnt w j^ 4TAjh Lemma 2.6. If P{IX is regularly varying with exponent then is slowly varying. Proof. Without loss of generality assume that Since is regularly varying with exponent +6 for all x> some Thus for all x > G(sx) 2tP{ t}dt t dt 2tP{ IX t}dt = G(sx G(sx  G(sx implying yjdy +6)(G(x)G(x that G(sx) = <. n  Taking the limit superior of both sides as x + w yields G(sx) m sup XE(x) X40 Likewise, for all x> G(sx) G(sx = G(sx ys}dy 6)2 y}dy = G(sx implying G(x that G(sx) G(x) G(x) 6)[1 Taking limit inferior s as x * in this last inequality yields .r G(sx) lim inf G(x) Since is arbitrary , the desired result follows from this and (23) converse to Lemma is false; for a counterexample Feller 288). Alb lhi nnn ma n rAiv ir.t7 see 6)[  i tim o rnn\i nt+ i r nrt T .i fC 4u I,. Tn n '%"i r Corollary 2.1. are i.i.d. random variables with , EX2 regular varying with exponent and ( are satisfied , then holds. Proof. Lemmas and 2.2, condition holds. In view Theorem it suffices to show that /B n n = o(1). Since is regularly varying with exponent we obtain via Lemma with = 2 that for all x > some Recalling we note that whence max there exists an integer so that max whenever So if we obtain n/ ak for all k = 1, Thus implies whenever Therefore when n is sufficiently large, 1 .  I n B R n k= a EXI( kc a. EXI( IXI ^L. 1 I I (since = 0) = 1 = m + m,, S *I,n . S 1B B = o( The following two corollaries will be shown to be immediate consequences of Theorem 2.4. Corollary 2.2. Let {X, be i random variables with = w and satisfying n max a be constants so that = O(s (26) mln Then  G(q ). ni hol ds. Proof. In view of Theorem 2.4, we only need to verify that prevail. max and observe aki' that n a > n s n max a 1 9 0 _ n max 2 2 and clearly implying that /n max For arbitrary , note that nH(q ) 2 nH(q ) 2 (since implies that slowly varying) (since via ( H(x)  G(x 3)). Thus nH(qn) Then, for all arbitrary and n sufficiently large max EB / ) 2 = nP{ xl E >  C n E n(cq C n 1 q E H(q) Cn E H( q ) Cn (q ) C n Cn E H(qn (;q )2P C n (29)) H(.n) thereby proving 10). prove in view of (27 we need only show that G(q ) n n ) max l ak 1 (30) Note that entails n = o(  G( ___ Utilizing the fact that G is nondecreasing we obtain 1 CnG ( < G(qn).  n 1 Thus (30) holds since G is slowly varying. Corollary 2.3. be i.i.d random variables with = o and satisfying be constants so that then akXk k k + N(0 where is defined as in Theorem 2.4. Proof. Again we will verify that and (10) are satisfied. Clearly, B n la k max 1 . 9 9 , n > * _ J 1. 1 *i * * max 1 mnn < G(inf  i  Recalling that is equivalent to G slowly varying we obtain G( sup employing 31). Hence obtains. Define max 1 and 8B n and note min that n a > n s 4 o. In light of (31 there exists constants * such that min max < C2, for all n > Recall that = nB2/s n and thus n n C1 Again using the fact B that n G() C2 2 is nondecreasing we B ) < G(qn) n n n < G().  C 1 obtain This together with slowly varying, ensures that for all E > G( sup k>1  tnf , n > From (32) it follows that C  2 Thus, for all e > )nG(q C2 2 c? 1 nG(q ) n whence = 0( Therefore for arbitrary EB / n "n ECs 2 nH(ea ) 2 Theorem does have a partial converse. however, impose relatively strong condition about the weights Remark. Let { be i random variables with Let {a a sequence of constants where = 0(n min Then implies and ( Proof. there exists a constant , such that which implies that n min n min for all n > Note that, definition of H . H(x as x Then, recalling G(x) whence Q(x) = x2/G(x) as x *+ O This, in turn, shows that 4 00 Therefore 2 = s G(q n n  and B n min [a j 1 B n max a 1 n  s n 2 a) 4  0 , n > = o( = o( then, employing Corollary 2.2. of Chow and Teicher 978) we can conclude that holds a,[H(EB,  (EXI( X So to establish note B / lak n k that for all e > for arbitrary max P 6B / n 6B / max 1 6B / = o( whence obtains. Then using we conclude that P{Ixl a2[H(B [11( / lak  (EXI( n /ak ))2 = p 40), min akH( B 2p nB P{ X n s2H(B n min min min a n min 1 1 n . mln la 1 (37)). min I Thus mmn min i (k = 0(1). ak ) Then for all x such that we observe that PiJX mmn S 1<(k since H(x)t) n min lak 1 mln ) =o0 recalling thereby establishing The next corollary an immediate consequence of Theorem and this last remark. It is the famous result of L6vy Khintchine 935) and Feller (1935) cited previously as Theorem 2.2. Corollary 2.4 (Chow and Teicher p. 300). are i.i.d. random variables with = 00 then  N(0 some and A iff ( holds ; moreover, be chosen while be taken W 00 = 4( Proof. Clearly, since Also ak EXI( = nEXI(IX ,n > whence the sufficiency portion of the corollary follows from Corollar 2.3. Necessity follows from the last remark since = n min 1 Properly Centered Central Limit Theorem We have seen that via Lemma , implies that e a,1 Then, we assume EX = 0 = 00. and P{jx regularly varying with exponent we showed in Corollary that holds. It is natural to ask what happens when the mean of X finite but not zero. we just shift the variables EX and achieve asymptotic normality? In other words, when does the CLT EX) hold? , n > are i.i.d. with arbitrary finite mean, while sequence are defined = XEX n EX, Again we suppose that = w and thus functions and G Likewise will , Q(x) be defined = x2/G(x) as in ( and (5). ,n > and B = Snqn n n ,n > Similarly we need to define analogous quantities terms of the random variable = EY2I( 2tP{ t}dt Also let Q (y) 1 /G (y 1 = Q1 ,n > Snq nfl //n, (44) Before we establish the relationship between these pairs functions sequences we need a few preliminary lemmas. Lemma 2.7 (Rosalsky, 981 ). If a( is the inverse of the continuous S,rTf \ 4 \ nn I ,F^ i f ntr ,tr nnl yar~ ^ ? I^  F*i 11 /tl ; hny *^ ~ n Allv l 1*^ ~rnH r; * 1r Proof. Rosalsky 1981), Lemma Lemma 2.8. Suppose that is slowly varying and EX2 = w. Then  H(t) as t + 0 and H is slowly varying. Proof. = EX and recall that Then, = X. recalling (42), for all t = EY I( = E(Xp)2I = EX2I(  2iEXI( i2 EI( xul for t since E t 1 via Lemma 2pEXI( Also EXi for t u2EI( P{IXi Therefore, since H(t)  as t + m (from we obtain H (t) 1 EX I( 4 4 } < 2 = ) For t EX2I = EX2I v+t) EX2I( = EX2I Also, = H(2t) for t EX2I = EX2 EX2 1t 2 1 < t 2 = EX I 1_ < t)  2 1 = H(t). 2 These two results entail for sufficiently large 1H( H(t) H(t) EX2I H(2t) H(t) and since is slowly vary we obtain EX2I  H(t). Tn vi P u n? (Uh'1 1" 'mi \ri e] re a u (  U(. ^a < *^m I(ut u+t) Finally for s > H (st)  H(st)  H(t (since (since  H(t)) is slowly varying ~ H (t) (since H (t And so HI is slowly varying. With this relationship between 1 and H established, we note that a similar relationship clearly must then exist between I and G when is slowly varying in view of Lemma 2.2. The natural question is whether Prior to establishing this we need to state facts about functions of slow variation. The following result (see, e.g., Feller, p. 282) characterizes the class of slowing varying functions and is known as the Karamata representation theorem. Theorem. function L(t) varies slowly iff it is of the form L(t) = a(t)exp{f e(ds s S (46) where and a(t + 0 c, as t This next lemma will demonstrate the value of the Karamata representation theorem and is a quite useful result.  H(t L(u(t))  L(v(t)). Proof. Since L(t) is slowly varying, by (46), L(t) t =a(t)exp{j dy}, 1 where E(t) + 0 and a(t) C 0 as t + Thus, L(u(t)) L(v(t)) u(t) a(u(t)) )xp a(v(t) ^dy y v(t) 1 1 E(y)dy} dy y < (1+o(1))exp{f max{u(t),v(t)} E(min{ut),dyt) min u(t),v(t)} Y since a(x) * c as x  ~ and u(t), v(t) wm as t 3 m. Using u(t)  v(t)  3 and e(y) * 0 we observe for all t sufficiently large that I(y) for all y min{u(t),v(t) }. Hence, via (47), for all large t L(u(t)) L(v(t)) max = (I+o(I)) min max{u(t),v(t)} < (1+o(1))exp{f dy} min{u(t),v(t)} u(t),v(t) u(t),v(t) {u(t) v(t)} = (1+(1))max v(t) u(t) = 1+o(1). Reversing the roles of u and v we obtain for all large t L(v(t)) L(u(t)) 1+o(1). Combining (48) and (49) we conclude that L(u(t))  L(v(t)). D Lemma 2.10. If EX2 = w and H is slowly varying, then B  B n Proof. Recall (42) and (43). By Lemmas 2.2 and 2.8, G(t)  H(t) ~ H (t)  G (t) whence Q(t) = t /G(t)  t2/G (t) = Q1(t) Let q(t) Q1(q1(t)) = t = Q1(t) and q1(t) = Q(q(t)) = Q1t), t Q?(t), t  Q1(q(t)) (since Q(t) 0. Then  Q1(t)), and Q(q1 (t)) By Lemma 2.7, L(t)  Q (q(t)). 1/2 Sft q (t) is slowly varying since Q (t) and G1 = t2/G1(t) is increasing (see Rosalsky, 1981, Lemma is slowly varying. Next, note that if u(t)  v(t) + ~ then, via Lemma 2.9, L(u(t))  L(v(t)). Applying this to (50) we see that L(Q (ql (t))) rn In t if1/2  L(Q1(q(t))) _ rn / + I /2 . n rn ( t \  f q1(Q1(q(t)))  ql(Q1(q(t Using q (t) Therefore, 1 = Q (t) recalling we obtain or qn *n and ( * n n  Snqn . 0 Lemma If EX2 = w and H B Smin a Sk is slowly  G( varying , then max max Proof. From Lemma we have slowly varying implies via Lemma 2.2, that G is slowly varying. Hence  w we obtain 1 Lemm a 2.9) ~ qn  G(  G( Lemmas min ]a and 2.2). Likewise max laki 1 max  G1 max 1 Thus, if G( n min a ak max then max 1 (52)). max Conversely if G max then 1 1 ak  GI ~ G(  G(  G.( ak) Lemma 2.12. If EX2 = , is slowly varying, = o( for all e > then for all e > Proof. Let E > Lemma there exists a constant such that n n for all n >  Recall that max + and so if n is sufficiently large, then for all k 2MIa k . ,n where = EX. Therefore EB / n (53)) [P{X + P{X L njv EB / n n Mja. = o( P{ X b Lemma If P then is likewise is regularly regularly varying varying with with exponent exponent Proof. Let L(t and L for t It need only be shown that since then, arbitrary  L(st)  L(t) (t). To this end, let 0 be arbitrary and let where = EX. Then whence L (t) L(t) P{wXr P xT ut Px X tl p {lx u+t t1_ 1+E + p{x 2 t (1+E) L() 1+E L(t) 2 = (1+e) (1+o(1)) since, by hypothesis, L is slowly varying. L(t) t+ (1+E)2 , and since E is arbitrary we obtain L (t) lim sup L(t) t**m Again, let 0 be arbitrary, and now let (1e) I/E, which implies that lint. Therefore L1(t) L(t) pt} + P{ P{ X > t t} + P(X lEC u+t) t 1} t PT X 1E) t L( 1E L(t) since is slowly varying. Hence lim inf L (t) L(t) whence L (t) L(t) t** This , together with shows that L (t)  L(t) which is tantamount to P regularly varying with exponent are now able to state prove a CLT for random variables centered about their mean. Theorem 2.5. Let { ,n > be i.i.d. random variables with = w and let P regularly varying with exponent be constants n min la i 1 eB / n with n max a. k for all E > Then EX) 3  G( = o( (1E)2 Proof. Via Lemma and regular variation of P we can conclude that and H are slowly varying. This , in turn, implies that 1 by Lemma 2.3. Let Y = XEX, and note that since = = we also have Lemma and the hypothesis that regularly varying with exponent we see that is also regularly varying with exponent Again Lemma we can conclude that and H 1 are also slowly varying. Next via Lemma we conclude that Now ( is equivalent min  G (B max Lemma Finally we note Lemma that implies EB / = 0( for all > O. Therefore, all the hypotheses of Corollary are satisfied for the sequence Hence ak k Then remembering that EX) n k=  ( B n d  N(O akYk k k N(0,1). = CO EX, 2.5 Asymptotic Negligibility In order to establish a CLT, "it is [generally] essential impose a hypothesis where individual terms in the sum S are 'negligible' in comparison with sum itself" (Chung 97). There are several different measures of negligibility. literature (LoBve, Laha and Rohatgi 979, p. 295) calls the double array ,n > uniformly asymptotically negligible .a.n.) max aX k k B n for all e > This condition is also known various other names, such infinitesimal (Chow and Teicher, 422) or holospoudic (Chung, 974, 98). Another measure of negligibility max aX Imedian( n = 0. (57) Fact. If {akXk *k k u.a.n., then holds. Proof. Let E > hypothesis there exists an integer such that I akXkn I 1 1 nfl4 n*o P+ .I 1 I. U The measure of negligibility that we are using our CLT the condition Under our hypotheses , it is equivalent akX k k P 4QO. Fact. If { ,n > are i.i.d. random variables then are equivalent. Proof. Let e > and define akXk k k Thus, akXk k~k n = P{) [ k=l k=1 n k=1 akXk eB ] aXk k k EB]} n akXk akXk Pnk U In n > a C A c. n A  max A  knl 1/*n"  /  w 1exp  exp{pnkI nk Pnk Thus, 1 HI Pnk = o( = o( Therefore (58) equivalent to (10) Our measure of smallness implies u.a.n. condition. Fact. holds then {ak k u.a.n. Proof. Clearly, for all k akXk k k max a.X. 3 3 EBj. Thus max P aX k k I> Bn n max a.X. J J3 I > Bn ' n = 0o( An Asymptotic Representation The CLTs in this chapter hinge upon hypotheses that involve func tions of regular variation. At first we assume that holds; later we suppose that a regularly varying function. In either case and H are slowly varying. We have seen via Lemma that a..  l~ I o.~... .., 1 SI ,n S> eBn '~ n * . ii r L for {Bn' > rl L.. J r V II random variables have a distribution which satisfies these hypotheses. To understand this class of random variables one needs to study the Karamata the behavior of regularly Representation varying Theorem functions. that all slowly We know, varying functions are of a specific form. Also, recall that a function is regularly varying with exponent if and only w(t) = tPL where L(t) a slowly varying function. With all this in mind we will now generate an interesting class of slowly varying functions which are applicable our CLTs with Suppose the distribution of the random variable satisfies some aexp 2t2 as t + .O Thus aexp t(log gSt)}dt t)1a  exp{ (log can check that that , for s > a slowly ~ G(x varying as x function Alternatively, observing one can easily verify that holds. For let a(t) E(t) = a(log Then mYn I a(log cx1 ) l MA = pYn I a1 i mll nil , G( We require that so that and we also need ensure that From we obtain = /n/G(qn n Since G is slowly varying, natural question is whether we can replace G(qn G(/n) and thus conclude that  /n G(n) thereby yielding via (7) an explicit asymptoti representation Define G = exp{ (log x) } This class of slowly varying functions some intriguing properties It will seen that holds Proposition 2.1. some If G but not all, = exp{(log a in ( , then for 0 Proof. (log We need  (log to observe Jn) the limiting From behavior we see that whence n*o /2, a< = T/niE = log = log 1 + log 2 1 + (log 2 = log /n and note that for all large n, q Hence from (60) n is sufficiently large, (log qn)a  (log /') = (log + (log 2 q ) ) nI  (log (log J/n 1 2* ^f + (log 2  (log /Jn)a  m = CX mean value theorem some 1 m m where Therefore (log q)a1 n )  (log a /n) 1 a a1 ) 1+o( +o(1 )m 2 Hence Cn /n) = (m /n)) a S2m > (m 2 Since is slowly varying we obtain for all 0 that G(x) for all large Then = nG(qn which shows 1 1El that for all 0 and large So for arbitrary and all large Therefore, recalling n is sufficiently large (log qn) /ha  (log = (log + (log 2 qn)a)t  (log (log + (log 2 nB/2)a)a  (log /n)a = (log 2 (log Jn)a)01  (log na = (m +m 2" cx K)m 2 mean value theorem some where m m< m + m m 2 Thus, (log qn)a n  (log ,/ a a=2 = 8 2a1 2)m This, in turn, yields for all 8 that /n)  m lim sup{(log n** qn)a  (log /) C in) 6alim m 2a1 Letting we obtain lim sup{(log n+ qna n  (log lim m 2a1 me01 This together with yields lim{(log qn)  (log lim m for 0 for a for 1 Therefore = lim (qn a n = lim( n ci I G (O/n) ax = lim(exp{(log qn)ac n0  (log npi A /n)a /n)a} /n/5 /n) Remark. In this last example we let G = exp{ but if let G shown  G a to be valid the conclusion of Proposition will now be for G Proof. Let G = exp (log and let G some function defined with 2 = x /G (x  G = x2/ Define Since have whence Utilizing Lemma with Q (x) 2 = x L(x) where L(x) see that is slowly varying. Applying thi s to ( obtain, via Lemma , that (t))] (q (t) a a  [Q (a "C (t))] 1/2 a "a Reapplying we can conclude that This , together with ~ Ga ar shows that (where for 0 (n)) a < n** = Q In view of ( and the example in Proposition interesting class question of distribution to raise s. The is whether answer prevails is affirmative for a large in view of the ensuing Propositions Proposition If G(x) and 2.3. + 0 , then G(x)  G(xxVf). These , in turn, are equivalent to (59). In such case, G is necessarily slowly varying. Proof. Recall that = /n/G(q ), n whence /n for n sufficiently large. Assuming that G(qn)  G(/n) and utili zing the fact that nondecreasing we obtain for n sufficiently large that G(/A / )) < G(/nj9(q ))  n = (1+o ))G(/n). Thus ~ (/in (cn)). Now to show G(x)  G(x/GTx) , apply Lemma twice. First we obtain G(/n)  G(V/nT) and then G(/c (In))  G(/n+1 (/n'T)). Hence, if /n /n'T we have G(An) G(x) G(x/GxT) G (VrWT) (/R/G/E) G(/n) G(qn = G(qn  G(^ G(/n) G(/in) G(c 1 T Next, we prove the sufficiency half of this proposition. Again we use Lemma assumption G(x)  G(x/GTx7) implying that x/G(Cx) (x/GTiT) Thus, Lemma G(x) x/G(Ti (x/GTxT) Therefore GCxV4ZiT)  CG( x/GTxT cG(x/Gdx7) whence replacing x/G(x7 with yields G(q ) *n = G(2n) 3)). Finally we will show that is slowly varying. then for suffici ently large G(x) G(sx) G(x/GTxT which, together with G(x)  G(x/G(x) shows that  G(sx). TI t^ I  G(  G( T "4 t% Grq ) n9, LL1 ^ .1. . _ I _ I Thus G(sx) ~ G(x as x + ~ for all s > and so G is slowly varying. With Proposition in mind we would like to find other conditions Before that we do imply we need , equivalently, to state G(x) a definition (x/ fxT prove preliminary lemma. that a nonnegative function defined preserves asymptotic equivalence at infinity  g(yn) whenever and {y , n are nonnegative sequences with + m We have already noted , in Lemma 2.9, that if L is slowly varying and a(t + CO then L(a(t))  L(b(t now establish the following lemma. Lemma a nonnegative , nondecreasing function defined on [ which preserves asymptotic equivalence at infinity. Let g defined on [0 agree with on the integers and be defined linear interpolation between the integers, i.e., g(x) = (g(n+1 g(n))(xn) + g(n for n < n+1, n > Then g(x)  g(x) as x S* and preserves asymptotic equivalence infinity. Proof. For all large x, writing n+1, it follows from hypotheses on g that , n > +0(1) g(n)+1  g(n+i1) _ g(n) g(n+1) g(n+1 g(n+1) g(n) +o(1 as x m, whence  g(x) as x + B. Then  yn  g(yn ) so g preserves asymptotic equivalence at infinity The next proposition, when combined with Proposition 2.2, and ( yields the explicit asymptotic representation (Vn). Proposition 2.3. Let G be defined as in ( suppose that G(x) * w. Then => (ii) => (iii  G(x/GCix) equivalently, (/n)) where (ii , (iii) are given other e exists + such that exp{ G(x)  "2" t(x) sequences G(u ) n with  G(v whenever and {v are real  log (iii) there exists , s(x + such that  0 ~ => G( 0 *; Moreover Proof. G(x) Suppose  G(x/ff(x) that implies holds that is slowly and let log ~ log varying.  w. = min{u nvn}  max{u Then n},'  n <   G(m ) n 1+o( b(mn) b(M ) n log M 2 1 log m log M 2  lo ))exp{ )exp{0O log M )log( o n )exp 0(1) log log v implying that  G(M Therefore, G(m ) n (M ) n G(u) G(v) n < _"  G(m ) n = 1+o(1 thereby proving => (ii) Next suppose that holds. It will be shown that (iii) obtains with = (lo Define = G(exp{x We will now verify that preserves asympto equivalence infinity. L  y  = and set = exp{ x v = exDy4 , , x > ~ = G(u ((log v = g(yn). Define on [0, as in ( 63). Lemma 2.14,  g(x) and g preserves asymptotic equivalence at infinity. Thus, exp{ (log x) G(x) exp{ (log exp{ (log g((l and to complete the proof it suffices to show that eventually nondecreasing. To this observe that suffices show for n sufficiently large and 0 that <1 t (n+t g(n+t ) Choose such that g(n+1)/g(n for all n > and 0 g(n+t) For 0 is differentiable on (t1 ,t and is right left rPqnoni uavl'v ( = g( ~ G(v entnn i nll nml /^n it +* < tl t A'(t) n g(n+t)  e n+t) '(n+t) n= [g(n+t) [g(n+t)]2 n+t g(n+t)2 (n+t)  (g  g(n+1)  g(n))] + g(n)] >[  [2g(n [g(n+t)]1 (n+1)] recalling the choice of N. mean value theorem guarantees existence a point in (t1 such that  A(t However and A'(t) n , whence (t2) n 2 n(tl thereby proving 64). Now, suppose that iii) holds. Then G(x/G(x)) G(x) exp{ s(xVG(x) /log(x/ iuy) r(x/G(x)) r(x) (1+o( exp og(x/~Tx) expr(x l r(x)  /log x J . log(x/G( i) = log x + log 2 G(x) = log x 2 log x r(x)  log Lemma 2.9) = [log implying (1+o(1)) for all large )/log x x that log( x/G(x) = +log x[1+ 2 r(x)/ log x /2 (since (since (since s(x) r(x)/log x r(x)/log x Jlog x + + r(x) /log x (since r(x) + x). Hence for all large hlogx(xx) fnr a1 1 1 2ra o r log s log + 0) /log x( = /log x  /log x x][1+ /log x( r(x)/loog x /log x( /log x Th >n fr nm G(x/ViT) G(x) +o(1) ))exp{ r (since r(x (since r(x thereby proving G(x)  G(x/Jcfi). Finally , the last assertion was proved in Proposition 2.2. 0 Examples We conclude this chapter with two examples illustrating some the results of the chapter. Example 2.1. Let { be i.i.d random variables with common density function f(x) Then /log k(X 4 N(0, Vrnlog n Proof. Note that EX = 2 and EX = 0. Also for x > Thus is regularly varying with exponent Then, G(x) 2tP{ t}dt 2tdt + 21og which a slowly = /log varying n, n> function. Then (log x)dx +log x)dx  nlog Note, that condition (ii) of Proposition is satisfied, whence  /nlog n. Next observe that max = jlog n. Set, for n > n Idt  n /(/n) n max la 1  /nlog n. Thus, recalling Lemma 2.9, G($n) = (1+o(1))G(/nlog n) = (1+o(1))log n = (1+o(1))G(/nlog n) = (1+o(1))G(8 n and so G(B ) n Finally, let P{Ix establishing (55). For n sufficiently large eBn/ akj P{ xl E/liog n 2/l og k since B  /nlog n) 4log k (by (56)) E n(log n 4nlog n e2n(log n)2 establishing (56). 2 E log n = 0(1) Then by Theorem 2.5, SN(0.11. log k(X  G(a This next example illustrates Corollary 2.1. It will also create a family of CLT since the weights are not explicitly defined. Example e 2.2. be i.i.d. random variables with common density function 2(log f(x) Let L any positive slowly varying and nondecreasing function. Then L(k)Xk * N(0 e/HL(n)log Proof. Firstly, integration parts, we verify that is indeed a density. f(x)dx = e2 2(log x)1d dx = e2[ x 3dx] , n > Clearly EX = 0 and EX2 = C. For x Se2 (log t)1dt  d = e(2(log t)1) t2) + f t3dt] e 2 = () log X and so P{I X is regularly varying with exponent e x G(x) = f2tdt + x2t(t) (log t)dt 0 e = (elog x)2 which is slowly varying. Note that since L is slowly varying, it follows that (see, e.g., Feller, 1971, p. 281) SEa k=1 L2(k)  nL (n). Also, G(x/G(x) ) = G(xelog = e (log(xelog x))2 P{Ix Then Proposition or 2.3  e/nL(n)log Let e > and note that n is sufficiently large VnL(n)log /. Hence for all large EBn/L(n)} n (since L(x) < nP{Ixl log(E /nlog E n(log whence obtains. It remains to show that holds. Note that min I 1 = L( max I 1 = L(n). Since slowly varying and B  eJAL(n Lemma yields Ge/nL(n)log /n' L(1) 1  G(/nL(n)log  s = o( P{lxl /nlog G(B / max ak, ) 1 ~ e/nL(n)log /n, G((n) L(n)  G(/nlog n). Let a = /nL(n)log n, n 1, and Sn = /nlog n, n establish (9) we need to show that G(a ) n  G(6 ). n Note that G(a ) n 2 2 = e (log(nL(n)log n)) 2 n)+(log L(n))+log2n)2 = e (2(log n)+(log L(n))+log2n) 2 e (log n)2 2 (by (3)) 2, 2 = e (log(/nlog n)) 2 1 2 = e (2(log n)+log2n)  e (log n) and so (9) obtains. n Then, via Corollary 2.1, n G(8B I CHAPTER THREE GENERALIZED STRONG LAWS OF LARGE NUMBERS Introduction In this chapter, we present generalized strong laws of large number (GSLLN) for weighted sums of random variables, Y) a.s. hypotheses of these theorems vary greatly In general, control the behavior of the random variables restricting the magnitude of the tail of the distribution Xn l Mos t of the assumptions that involve sequences , n > only depend on the absolute value of their ratio. thus proves convenient to define sequence This notation will used throughout this chapter. sequence will for the most part be the null TQ cha nt hn rromi n na1 , n > n/C!i (Y Q Q ^ r.ri 1 1 caom lon^Q P on ip m n n/d Preliminary Lemmas The theorems in this chapter assume that the random variables are either that i. i.d. sequence or stochastically is stochasticall dominated. y dominated there exists a constant o such that < DP{ IX t/D} Sn > It is important to note that condition does not pi restrictions on the joint distributions of the random variables Also it should be clear that if D satisfies then number larger than also satisfies Finally note that if the random variables are i.i.d., then holds with = X and D = 1. Lemma Let X and X be random variables such that stochastically dominated in the sense that there exists constant * such that t/Do} Then for all t for all q+1 D E s/D ) 0 s/Do. 0 Proof. Note that + EIX t}dt q1(Ix Lemma t/D }dt OJ = qD t}dt [sqp + ED Lemma s/Do} s/Do) 0 Lemma will be used in establishing Lemma which, as will become apparent plays a maj role in this chapter. Lemma Let {X ,n > and X be random variables such that is stochastically dominated be constants with max 1 where = 0(n) some + Dq tqp X qI sqP{ where as in ( Then for all 0 1q k ^qEIk < Mc  k Proof. Let 0 let D = max D,/M} max c and set do = 0. Note that and (5 ensures that view of (5) there exists a constant w such that q z nj =n for all n > Note that so ( holds with and then 1EI Xk ^q k < Mc )  k q1(Ixk 1 q+ qDo q o ck k EIXlI( X oCk) ok SD2q+1 + D 0 oCk0] Lemm a with series in the second term of (9) converges since it is bounded above 6)). D2q+ o The series in the first term of (9) is majorized 1 El A o n1 < Dd )  on k q k=n c k dn1 o n1 nP{D = C 2: n= n=1 d }Ddn o n o n odn1 o n1 qI(D o n1 qI(D Dodk1 thereby proving the lemma. As previously noted, if the sequence is identically distributed then automatic but this is hardly necessary. is quite eas y to show that if the sequence , n > belongs scale family then holds (subject an additional assumption) follows. (The symbol denotes that the two random variables have same distribution.) Remark. Let X , 0 Then is stochastically dominated Proof. Let D  max{ 1 ,sup a ). Then for all n >  Pr 1 I A I f /n wellknown example a scale family is the class mean zero normal random variables. Equation distribution compares of the random the magnitude variables and of the tail the constant of the s. It is well known (see, Chow and Teicher 978, 8990) that whenever w (strictly) every random variables where a strictly monotone extension of { Hence , in some situations the question as to whether or not holds is immediate since, for example, from we obtain Finally , we need to comment on condition It is clear that is of the form = n some , then obtains The question arbitrary The next at hand three is when lemmas does address hold this question. Lemma If 0 + and then E I k=n p if {c = O(n/cn , n > eCp S 1 k=n C = O(n/c ). n Proof. Note 1. that + and ensure that Therefore   then k=n c * 1< qp 1 qp k=n c k = O(n/c ) qp n n Lemm a = O(n/c ) If 0 = 0(n) then e and ( holds with same Proof. Since = 0(n) we obtain whence Clearly + and  (q b = f(l n la . r 1 1 qp ck 1 qp' O(n/c ), n some mnnv  0, = 0(n) thereby proving the lemma Lemma 3.5. Suppose cq/n n for some Then holds rn lim inf  C some integer Proof. Let d = o/n, n Then + and So ( is equivalent lim inf n+r some integer This , in turn, is equivalent rn lim inf  d n** n some integer On the other hand, is equivalent 1._ k n kdk k=n k 1 = 0( ). d n Hence we need only show that whenever /q(d /qd( is equivalent rn lim inf  d n*wo n some integer This equivalence was proved Martikai nen 985) following lemma is quite useful our work. From it is clear that but if c /n n + for some then can be weakened. Lemm a Let X random variable and let constants such that c /n n some Then either for all A or else for all 1 Proof. See Stout a new simple proof Rosals (1985) Thus , if 0 c /n n + for some then in order to verify we need only check that see . 1 n n + for _L _ Generalized Strong Laws of Large Numbers for Weighted Sums of Stochastically Dominated Random Variables With these preliminaries accounted the first maj or result of this ch indicators apter may no are present w be established. It is unfortunate of the conclusion that of Theorem However , under additional conditions, it is shown that holds where n do not involve indicator functions. Theorem Let {X ,n > and X be random variables such that stochastically dominated in the sense that there exists constants such that t/D2} , n > ,n > and I be constants satisfying and ( with = 2, where is defined some constant then every the GSLLN uk a.s. iYr I / t  ftir T  J I _  _1 = O(n) then v. a.s. where = EX 1 and = E{X n IX', n1 ' Proof. with Let M > = max{D,1 The hypotheses whence ensure Lemma that and ( obtains hold with = 2. Let Z Mc ) n Define ,n > Observe that for k measurable and h ence Ik1 ak   bk a.s. Thus This a f Ir b n nUn) n n is equivalent ,n > to saying is a martingale that difference sequence. k) I'n' martingale. It will now be shown that for i S. t ,X = o(Xo, ak bk k S .tX , n > , k Observe that E(Z..i )(Zj .y ) 11 i J i (since Z.y. is measurable) = E{(Z)E1 Z  = 0. Next, it will be shown that n a k=E (Z  k=1 k converges a.s. In view of the martingale convergence theorem (see, e.g., Breiman, 1968, p. 89) it suffices to show that lim sup El E Zk I n*o= k=( k Note that E{(Zk )2 =E{Z l~k 2p, E{Z, k 2 + k (since Uk is ak measurable) k1 = Efz2I i 2  u. = E{E{(Z ui)(Zjj) }} 1 1 J J J~ Thus n a ))2 lim sup(EI b (Z k) ) no k=1I k na2 < sup E( Z b(Zk )) n>1 k=1 k = sup{E n>1 n an a n a. a. S(k)2(Zkkk)2 + 2EZ C(t) ( ()(Zj ) k=1 bk b.j j n = sup{ n>1 k=1 2E(Zk k2 n n a.a. + 2 E(b bj)E(Zii)(Zjuj)} i (by (19)) EL E(Zk Ik2 =1 c SLE{E{(z )2 =1 ck 2 k 1 =1 Ck r E7Z k k k 1 2 EX I( 2 k < MCk  k as already noted , whence obtains thereby establishing 20). Using and the Kronecker lemma, we conclude a.s. However, k) and the second term a.s . via b  and the BorelCantelli lemma since (since and ( I 4)). under Cn for all n > whence for all k CD D3n 2 3 < D1  1 CD n 3 4)). Therefore e via ( we obtain w and E This in turn guarantees that the conditional expectations all exist. = inf Note that 0 and ,n > Under some Utilizing Lemma we obtain No ) n = Nc P{ IX > t}dt Nc P Nc_/D } + Df t/D2}dt (13)) P{ID t}dt] ' I I(ID Nc ). E E X I( X SC n n n=1 n nO Nc ) n SD1 Z EjD2XI( D2X n=1 n Ncn ) n < D1 Z1 EID 2X( D2X 1 n=1 n Ne ) n 1 c n=1 n k=n EID2X II(Nek ID2XI < Nek+ ) ~ k+1 since EIDXI I(Nek (D1  1 Nek+ P{Nek ID X I ID/X < Ne + < Nek +l  ^k+1 1 I) C n=1 n k+1 n=1 n < D N S P{Nek 1 k=1 IDX 2 < Nek+l }Ck+ k+1 nl n n=1 n (since ek < DIN  1 P{Nek ID2X < Ne }C(k+1)  k+1 (17)) < ck) < Nek  k+1 < 2D N = C < Nek+l  k +1 < Nekl  k +1 N en} Men } (since Mc ) and ( Hence 4)). we obtain E C cn n Nc ) Next , it will be shown that Using and the fact that M > D2D 2 we obtain Ej C nlI(Mc No ) P{Mc < ND 1 {Ilx Nc} Mc_/D} < ND  establishing 23). Combining and ( yields 1 SEIX n Mc ) n SEIX n I(Mc n X Nc ) n 1E X EJX Nc ) n Hence the BeppoLevi theorem, I (lx E E{I X I( X  > nl n nn 1 E{E{ X I(Xn n=1 n Z IEX II( X > n=1 n Men ) > Mcn) }} n n Me ) n This implies that Slan n bn In IXn n=1 n a.s. Then by Jensen's inequality for conditional expectations and the Kronecker lemma we conclude that n I E ak (ik k) k=1 E a (E{X I( Xkl k=1  E{Xk)I < Mck) } k} akE {XkI ( Xk I k Lk 'k Me ) n n1 MCk k I Therefore, n recalling (15), n v) ak(Xk k) ak kk) a.s. thereby proving (18).D Remark. Note that condition (16) is automatic Furthermore, + for some a > 0 then, via Lemma 3.6, P{IXI Consequently, for all D2A and hence M can be chosen arbitrarily small and holds for all M This first corollary is a well known SLLN for sums of i.i.d. random variables and is essentially due to Feller (1946). It is extension of the MarcinkiewiczZygmund SLLN to more general norming constants. Corollary 3.1 (Feller, 1946) Let {X, be i.i.d. random variables and let be positive constants. Suppose that either = O(b (ii) L J h /n +. h /na + for some a c /na P{ IXI then Z X k k=l b n a. s. Proof. Clearly (13) Defining a = 1, holds with D we obtain and X , via (2), = X . that c Thus (24) is equivalent to (14) where D3 = 1. Suppose (ii) or (iii) hold. Then b + 0 and b /n n + for some Therefore /(rn)A b /n8 n for all integers r Hence, in particular, 21/2 b2n lim inf  n* w bn n**0 n 2 This together with b /n + shows that, 2 1 n k=n b k 1/2 2via Lemma 35, via Lemma 3.5, = 0(n). = D2 