Ding Shisun
Dept. of Mathematics
Peking University
Beijing China
Prof. K. L. Chung
Dept. of Mathematics
Stanford University
Stanford. California
94305
U.S.A.
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Draft of a statement on the life of P. L. Hsu
TaoLu Hsu was born in Peking in 1909 and obtained his
B.Sc. degree from Tsing Hua University in 1933. He spent the
years 19361940 at University College London where he received
a Ph.D. in 1938 and an Sc.D. in 1940. From there he returned
to China to accept a Professorship in the Department of Mathematics
at Peking University.
The war years were difficult. Hsu continued his research
while living in a cave; letters from 1943/44 to Professor
Neyman in Berkeley mention starvation. Efforts to bring him
to the United States 'finally succeeded, and in 1945 he arrived
just in time for the First Berkeley Symposium on Probability
and Statistics. A semester of teaching at the University of
California was followed by a semester at Columbia. This was
the year in which Hotelling moved from Columbia to North
Carolina and he offered Hsu an Associate Professorship in the
Department he was creating there. Thus Hsu spent 1946/47 in
Chapel Hill from where in spite of many efforts to keep him
in America he returned to his Professorship at Peking
University.
Professor H. F. Tuan. of Peking University informs us.
that 1&4died in his house on the campus of Peking University
on Dec. 18, 1970 from chronic tuberculosis. There was a
memorial meeting in his honor.
Hsu is affectionately remembered by many students and
colleagues as a gentle, shy and modest man who was reticent
about his personal life, but had a strong influence as a teacher
and model of a scientist. Isadore Blumen, who was a student
of Hsu during his year in Chapel Hill, writes: "It was Hsu's
insistence on simplicity combined with depth of understanding,
clarity without avoidance of difficulty, and above all a deep
and obvious but unspoken commitment to the highest goals and
standards of scholarship which attracted us to him." Ralph
Bradley remembers Hsu's lectures as "models for the future"
2
and t Robbins says "he was unforgettable and I fear irreplaceable."
Hsu's statistical work was concerned primarily with inference
in univariate and multivariate linear models and with the associated
distribution theory, both exact and asymptotic. A more detailed
discussion of this and his probabilistic work follows.
Department of Mathematics,.
Peking University,
Peking,
China.
5, July 1976.
Mr. Erich L. Lehmann,
Department of Statistics,
University of California,
BerkeleyCalifornia 94720,
U. S. A.
Dear Mr. Lehmann:
I received your letter of March 4, 1976. We appreciate
that at the request of the Institute of Mathematical Statistics,
you are preparing an obitunary article about Professor P. L. Hsu.
Professor Hsu died of a chronic tuberculosis in the early
morning of December 18, 1970 at the age of sixtyone years in his
house on the campus of Peking University. Not long before his
death, he was still doing researches on combinatorial Mathema
tics. We held a memorial meeting in his honor.
Enclosed please find a complete list of research papers
published by Professor Hsu and a brief account of the academic
degrees and positions held by him.
Very sincerely yours,
HsioFu Tuan,
Chairman.
PaoLu Hsu (19091970)
B. So., Tsing Hua University, 1933.
Dr. Sc., London University, 1940.1
Professor, Department of Mathematics, Peking University, 19401970
(on leave, 19441947).
Visiting Professor, Department of Mathematical Statistics, Columbia
University, 1946.
Associate Professor, Department of Mathematical Statistics, University
of North Carolina, 19461947.'
Member, Department of IMthematics, Physics and Chemistry, Academia Sinica,
19551966.
Member, Editorial Cormitte, Acta Mathematica Sinica, 19501966.
Member, Editorial Comindtte, Progress in Mathematics, 19551958, 19621966.
Member, The Fourth National Committe of the Chinese Peoples' Political
Consultative Conference, 19641970.
Papers by PaoLu Hsu:
1. On the limit of a sequence of point sets. Bull. Amer. Math.
Soc., 41 (1935), 502504.
2. Contributions to the twosample problem and the theory of
the "Student's" Ttest. Stat. Res. Mem., 2 (1938), 124.
3. On the best quadratic estimate of the variance. Stat. Res.
Men., 2 (1938), 91104.
4. Notes on Hotelling's generalized T. Ann. Math. Statistics,
9 (1938), 231243.
5. A hew proof of the joint product moment distribution. Pro.
Cambridge Philos. Soc., (2), 6 (1940), 185189.
6. On the distribution of roots of certain determinantal equa
tions. Ann. Eugenics, 9 (1939), 250258.
7. On iterated limits. J. Chinese Iath. Soc., 2 (1940), 40
63.
8. An algebraic derivation of the distribution of rectangular
coordinates. Proc. Edinburgh Math. Soc., (2), 6 (1940),
185189.
9. On generalized analysis of variance.. Biometrika, 31 (1940),
221237.
10.On the limiting distribution of roots of a determinantal
equation. J. London Math. Soc., 16 (1941), 183194.
11l.The limiting distribution of the canonical correlations.
Biometrika, 32 (1941), 3845.
12.Analysis of variance from the power function standpoint.
Biometrika, 32 (1941), 6269.
13.Canonical reduction of the general regression problem.
Ann. Eugenics, 11 (1941), 4246.
14. On the problem of rank and the limiting distribution of
Fischer's Stest function. Ann. Eugenics, 11 (1941),
3941.
15. The limiting distribution of a general class of statis
tics. Acad. Sinica Science Record, 1 (1942), 3741.
16. Some simple facts about the separation of degrees of
freedom in factorial experiments. Sankhya, 6 (1943),
253254.
17. The approximate distribution of the mean and variance of
a sample of independent variables. Ann. Math. Statistics,
16 (1945), 1291
18. On the approximate distribution of ratios. Ann. Math.
Statistics, 16 (1945), 204210.
19. On the power functions for the Etestand the Ttest. Ann.
Math. Statistics, 16 (1945), 278286.
20. On a factorization of pseudoorthogonal matrices. Quart.
J. Math. Oxford Ser. 17 (1946), 162165.
21. Sur un theoreme de probability denombrables. C. R. Acad.
Sci. Paris, (1946), 467469. (with KaiLai Chung)
22. On the asymptotic distribution of certain statistics used
in testing the independence between successive observa
tions from a normal population. Ann. Math. Statistics,
17 (1946), 350354.
23. Complete convergence and the law of large numbers. Proc.
Kat. Acad. Sci. U. S. A., 33 (1947), 2531. (with H.
Robbins)
24. The limiting distributions of functions of sample means
and application to testing hypotheses. Proceeding of the
Berkeley Symposium on Mathematical Statistics and Probabi
lity, 1946, 359402. Univ. of California Press, Berkeley
and Los Angles, 1949.
25. Absolute moments and characteristic function. J. Chinese
Math. Soc. (rew Series), 1 (1951), 257280. (in English
with abstract in Chinese)
26. On characteristic functions which coincide in the neighbour
hood of zero. Acta MIathematica Sinica (Succeeding J.
Chinese Math. Soc. (New Series)), 4 (1954), 2132. (in
Chinese with abstract in English)
27. On a kind of transformations of matrices. Acta Mathematics
Sinica, 5 (1955), 333346. (in Chinese with abstract in
English)
28. On a kind of transformations of matric pairs. Acta Scientia
rum Eaturalium Universitatis Pekinensis, 1 (1955), 116.
(in Chinese with abstract in English)
29. On the simultaneous transformation of a hermitian matrix
and a symmetric or skewsymmetric matrix. Acta Scientia
rum Naturalium Universitatis Pekinensis, 3 (1957), 167
210. (in chinese with abstract in English)
30. The differentiability of the probability transition func
tion of a purely discontinuous stationary MIarkhoff process
on the euclidean space. Acta Scientiarum Maturalium Uni
versitatis Pekinensis, 4 (1958), 257270. (in Chinese with
abstract in Ehglish)
31. The absolute continuity of the distribution function in the
class L. Acta Scientiarum Iaturalium Universitatis Pekinen
sis, 4 (1958), 145150. (in Chinese with abstract in English)
32. An association scheme M3(6) which is not a L3scheme. Acta
I'athematika Sinica, 14 (1964), 177178. (in Chinese)
Papers by Bann Cheng as representative for a seminar conducted by
PaoLu Hsu:
1. The limiting distribution of order statistics. Acta Mathe
matica Sinica, 14 (1964), 694714. (in Chinese)
2., Partially balanced incomplete block designs. Progress in
L.athematics, 7 (1964), 240281. (in Chinese)
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A General Weak Limit Theorem for Independent Distributions.
by
P. L. Hau
1. Introduction. For every integer n let there be n
independent random variables 1)(R. V.): X,. (j = 1, 2, ...,n).
The distribution function (D. F.) and the characteristic funotion(C.F.)
of X_ will be denoted by F .(x) and f (t) 'X \\ )
respectively, so that
The D. F. ?n(x) and the 0. F. fn(t) of the sum X., + X, +...+ X
are then, respectively,
(1 2) F,,(i)= Fn,,(i) *F tx) F W F)
where the star denotes convolution. The weak limit problem in
its full generality requires to find conditions necessary and
sufficient under which
(1.3) R?. "^ t') zFix) atl cwJj f Fn
where F(x) is some given D. F. This is equivalent to the
following:;)
(1.3')
L fj(t ) f(t),
,,n:5 OD
llandom variables are used in this paper for convenience
rather than necessity. The weak law in the theory of probabi
lity being purely functiontheoretic, the concept of random
variables may be dispensed with.
( ) denotes the probability of the relation inside ( );
E denotes the mathematical expectation.
3)1;v~k tzj
where f(t) is the 0. F. of F(x). In a highly elaborate paper
Feller [4) solved the important case where F(x) is the Gaussian
D. F., under the following assumption:
and in the nonessentially restricted form F .(x) = V. a.x).
Gnedenko 23 and Maroinkiewiozs investigated the general
problem, and in the latter, with the aid of Khintohin's
results K proved that under the assumption (0) the limit law
T(x) is necessarily infinitely divisible (Paul Lbvy !t, Khint
chinf' ). lMarcinkiewicz [W] also dealt with the case where
the limit f(t) in (1.3') is an integral function of a complex
variable, and obtained explicit conditions necessary and
sufficient for (1.3) for the cases where F(x) is the Gaussian
law, the Poisson law, and the D. F. of the "sure number" zero.
(The last case being the case of the weak law of large numbers. )
In this paper we shall also take (0) as our assumption and
shall solve the problem for the general case where F(x) is an
infinitely divisible law, specified by the fact that its C. F.
f(t) is the following one:5)
(1.4)
where G(x) is a nondecreasing function with G() G( ) < c.
4 umber in square brackets indicates the references cited
at end of paper.
54his form is due to KhintchineT.
2. Statement of the theorem.
THEOREM. Let F (x) be defined in (1.2), f(t) be the
function (1.4) and let F(x) be the D. F. having f(t) as the
0. F. Let condition (0) hold. Then in order to have
(2.1) , Fc o F at all continuity points of F(x),
it is necessary and sufficient that the following relations
hold at every x > 0 such that + x are continuity points of
G(y)s
V16
The proof will be given in the next two sections. In
order to have a condition on fnj(t) which is equivalent to
(0), we prove the following lemma in spite of the fact that
the proof may be found in Feller's paper [l].
LEMMA 2.1. Condition (0) is equivalent to the following
conditions
( o<) lim Max I fnCt o uniformly in every finite
interval.
PROOF. We have
I 1A i 6 J (A) E i j
Hence
Max lf)gt \) J ^
for every 2 7 0. If (0) is true, then it is easy to deduce ( o)
from the last inequality.
Suppose next that (o) is true. Then, since
00
we can find N = N(x,s) so that
Integrating both sides with respect to t on the interval (0, 2x1 )
we get \5. "
Ijx I, X/^
whence
Hence (0) is true. q.e.d.
By virtue of (o), log f ,(t) is defined in every finite
interval of t, for all sufficiently large n and all j 4 n, if
we fix log f .(o) = 0. 7'e then have the following simple lemmat
LE1!':A 2.2. (0) and (1.3) are equivalent to (o<) and ( ),
here ( ) is the following condition:
, /+A ) X
This follows directly from (1.3') on taking (1.4) for f(t) and
taking logarithms.
3. Proof of the sufficient part of the theorem. 'e shall
call any positive number a a continuity number if all the F,
and G are continuous at both a and a. Throughout the following
pages we use e to denote any, not necessarily the same, quantity
such that lei 6 1. We write
I,
I 1 ') (
6)h
131^
In this section our hypotheses are (0) (or its equivalent (os),
(I), (II), (III) and it is required to prove (3). We proceed
with a series of Lemmas.
LEMMA 3.1. We have
(3.1) 1 c. (x) 0 o for every x > 0.
PROOF.
Using (0) we get X M'
whence the result. q.e.d.
LEMMA 3.2. (I), (II), (III) imply the following:
(II') A x)
(3I.) W C,() ) oey,
for all x>O such that both x and x are continuity points of
G(y).
PROOF. The only thing new is (III). To Drove it 'e have
I. 1
2 Cr (x) +C C(x)
J 0 I
oo
Bn(^) ^.H.
Is1y L*
9
!?k
W
'7 Crjh
R~1(a
J
L{ 1F:19 ) c ) (x) A ,4 x J. .
J J
ance the result follows from (3.1),
Se. 0( .
LE A 3.3. Let E be any positive number, and let
hen
(i)
S I'w
91900
?1,
Tf is a continuity number, then
is bounded in n for every
fixed t.
V nOOF. 'or (i),
_u ol;cws(E)j tf)
V.,' e the result follows frorc (o<) and (3.1).
;ix ;if <,(,())
;d P,
ej J'~i JI J
I~k~
tj
Ixl Is
41I I 3. A 6A a)
I~I'~.
t C,. JdR ) o t 8, J I t1 ^ a
iX) E
f.t) I.
For (ii),
IJ tI
1e360
(I) :nd (II).
(t; = t e
if c (6 )
liCY V&)
tex^^
= ^ ^/U Lot 6., +^ ,^
whence the result follows from (3.1), (IP) and (II').
LEMMA 3.4. Let be any continuity number. Then
(3.2) hL ____
PROOF. For the real part
A (x) is a nonincreasing function of bounded total variation
A (L) which is also bounded in n, by (I'). Hence
For the imaginary part Let jbe any continuity number. Then
jj
7or Z x < however,
Thus \On(x)\ is bounded in both x and n, because On() and An()
are convergent sequences. Hence enotingeither the upper
or the lower limit,
TJ\Ai~d.FnI
) J 0
17
e
Since A(i) 0
,Lt
)I.
as . o, we get
JiQAILX
Proof of the sufficiency
of the conditions.
Let be any
continuity number. we have
z{j ( ;; ) =6E
d Fd (x) +
 l 3 3
 ?:jfi+ (xy t2 (XcC.+ 16f l d9F,/
Ijl<
)d^ T P H
3. \>t.#
c.d c,) ix
e
Ii,
I
i
Crx) d t 0 Xt
C~d^ e~x
" cx).
q.e.d.
IXI z
4T
(e
.
14 t0V
,^ ,) +6M)l
< xlu
1 11< It
= Ct).A(g) ^^t.Lt^^ t) + ^pj tc+ c)+ re'^)6F tc4)rA)
4 Oin() A 4 2tc(0 )A n(t) + ( c(t)
(xl~1
Hence, using (II'), (II), (3.2) and (3.1) we get
(3.3)
A; (t ) (
1 0 t3
I(le .
14>
Buit, for n sufficienQr large so that
j 4 n (of. Lemma 3.p(i)),
I \ (t J for all
I' f;(~~)j
 t~) =
~i
i M 0 +
and so
Using (III), (3.3), and Lemma 3.3 we get
/ a#x If~
dp~) t~(~)
As g, tends to 0, the term involving e tends to 0, Hence
I i ef
;x d^dq(3)
M. i+^
(f'E) j
It ,)^h ,i C* I / 1) J A
oa
which gives (). Hence the sufficiency part of the theorem
is proved.
4. Proof of the necessity part of the theorem. We first
together with their proofs, may be
found in Apaper [1I. We give their proofs here partly because
in Feller's paper he does not state these lemmas separately,
which makes it hard to refer to, and partly because Feller
restricts (nonessentially) himself to the form ~ () a(A.x).
;o rhmem1er th.t (OX (or its equivalent (\ )) ( \) and (A.1)
11a
In Lemmas 4.1 4.4 we consider a system of D. F.'s FI n(x)
together with their 0. F.'s f' (t) ( (t) + n(t)
(j = 1, 2, ..., n; n = 1, 2, 3, ... ) satisfying the following
three conditions:
(0d) YL M1A I[fI() =o uniformly in every finite interval
F I I fn V '. (t)
(M ) 0 is a median of every F' nj i.e. F nj(*O) 2 9,
1 F'n(0) < .
Throughout Leinmas 4.1 4.4 it is understood that (o'), ( ) and
( f) are satisfied.
LEMIMIA 4.1 We have
() C (I t).) for 1Si1 ,
roof: We need only consider the case t > 0. Let J1 be
the set of values x for which sin tx ? 0 and J" be the set
of values x for which sin tx 4: 0. Then
13
(4.2)
J+
.J
7"7
By Schwarz inequality we have, if J denotes either J' or J",
J.
(4.3)
C j" A
,j,,^ ^ cP. ".ff^^0^ c^
_y ( ^/ '*?
21d .'ijj"~dpy
Gince Iti T, J" contains the interval 1oj] Hence
T
 hF co) +J F. 4 F
T^^ 7 17,
AGimilarly, because J* contains the interval Tor we have
J F J I F ( o) J : .
we can find N,(T) so that
I 4<
i,1 J
for nO N (T),
j~ n.
oence for J equal to either J' or J" we have
substituting into (4.3) and using (4.2) we get (4.1), with r..d
 1 0.q.o.d.
LEMMA 4.2. We have
cFq
A
T{
\ /I
If
/V (T)'
(4.4) i , W) CT) Iiz'r.
PROOF. By Lemma 4.1 we have
i (T,, *r) ML U j ?^ Cjs )llt ) (tlT, '1 2u/i), ,L.
Dy (ck') (which is equivalent to (0), we have 1 V, < im= ,
whence Su, > for Itl 4 T, n = N2(T), J a n. Then
log(>u 2) V L4 t(u
whence
(4.5) u', 1 j It If i V 177), _y /'inT
3 fn
By ( ) the right side of (4.5) tends to a limit which is a func
tion bounded in t for itl 6 T. q.e.d.
LEMMA 4.3. We have, for all n,
(4.6) S f ()) f XL')?
PROOF. If is given, we have, by Lemma 4.23,
whence
f "t1( ) (t1
I
'1.
Integrating with respect to t on the interval (0, 2 ) we get,
as in the proof of Lemma 2.1,
SKIK7Z
which is the first assertion in (4.6). To prove the second
assertion let be given and let c = "(Z) be so determined that
1 aoxt x > ' c' for all Ix\ < Then, by (4.4),
3
o 0
H ^t y c.x)F (7
S (d I
j x( =
~x)
1;,
N(t) so arg ta
) 9
q.e.d.
LMMA 4.. We have
,, (t) t
j ~
~I'~JL
jfA~.zt~ I)
cv>)
v~
I1l >
z dR, r)Y
2 i. i
j 
,,^7I
7.e/.
(4 )
(A I
PROOF.
= o0
')I l
;C 0o
d
(~
/
2?. ~
SJS .
i
C14,
.2 f xy
I?i
(J^Gt.n/ I)
d w
L d ) j
15
Using ( oa) and Lemma (4.3) we get E n'" 0. Also,
zF,"+ MAP ==j 16 1 a j^ 3 t K O \ wl? i [dY
4H( H) C"^"*) f/
Using (0) and Lemma (4.3) we get 3l i  dt ,2(x) ,
whence E 0. q.e.d,
(4.3W XLA.*'A I\ 7. )
PROOF, As is ea y to ri:
(4.A) are respeotivel the r
and the imaginary part f the r
and V(t). Then, by (
(4.11) 11 ) 1
15a
Now we return to our original {F and f *. Our hypotheses
are now (o) (or its equivalent (0)) and ( ). Let yL be a
median of Fl. i.e. F(7nj.o) 4 and 1 F(/,1) 4 Let
,, = x, ,( X then F' (x) P (x'. t x)= Fi (x +lPy V),
f, (t) f *(t)e')A/",'. By definition tF,j satisfies (M).
Also ((3 ) implies that f. satisfies ( '). We shall prove
that ff. satisfies (oG'). This follows from the following
lemma:
LEMMA 4.5.
(4.8) lim Max ?ni = o.
4l3oo ^.^3
Proof. By (0) we have, for any > 0
J e < 1 for 7 2 ), ^
ixl>P
This implies that J"/ t e for n N(z), j < n. q.e.d.
That f, satisfies (< ) then follows as in the proof
of Lemma 3.3(i). Since all the conditions (<'), (C) and (M)
are satisfied, we shall make use of the Lemmas 4.3, 4.3 and 4.4.
We write
(4.9) A/ /V'1,+ F 3>, C. 
LEMMA 4.6. We have
(4.10) r 3 y =) for every x 0
4 o
Proof. As is easy to verify, the right sides of (4.10)
15b
and (4.11) are respectively the real part (with the sign
changed) and the imaginary part of the right side of ( ).
Call them U(t) and V(t). Then, by ( )
(4.12)
1h 30 C4^ i'usu ^a
t being fixed, we can have N so that
i '(t) e 
On account of ( ). Since
we have
, for n N, j n
I ( /
It4,~ CLI ~ (IuR~ .~. LI
 l IA^' V + Cl u()2 d (,UJ)436I/ 'f41 ( 3...u)
S/2.
(A hi^ ^ i L

.1. u
= j" /,, /
c :LZ. + 4 1i /ih I(Bu )
^ 2. ^ i yi ^f
By (o<) and Lemma 4.2, the term involving G tends to zero.
Combining (4.12) and (4.12.) we get
(4.1 ')
'L'. ,
Thus (4.jir) is proved. Using (4.V) on the first equation in
(4.1 ) we get (4.V). q.e.d.
LENMA 4.7. We have
(4.1k)
u It) I f I
S.)) av)t tb), for every Xo
?C1 I
l e at tblj *c ,
where a(x), b(x), a, b, c are constants, the first two defending
on x.
, 
v v t T
Hence
(4.13)
. / (A ^
"t
LrIV
(7 "
= Vat).
PROOF.
(4.1 o,,  IV). )iW.y 2. i
(4.1V) now follows from (4.1) and e si (46).
Also,
whence, putting t s 1,
(4.,, :4J/.; > ,
Sino tends to a limit by (4.V), the right side of
(4.lV) is bounded. Hence
>yde j a constant.
Tlen, (4.1'V) gives
( / /
,hich implies (4.16), by Lemma 4.6. q.e.d.
Throughout the following pages we shall adopt the following
notation
isotioe that g(t) is the 0. F. of the uniform distribution on the
interval (1, 1). '"e shall use TU to denote a R. V. independent
of all the X,; and whose distribution is the convolution of
three uniform distributions on the interval (1, 1). It follows
that g3(st) is the 0. F. of tC g3(st)f (t) is the 0. F. of
TLr + X,. and yI) 3. The D. F. of Lr will be denoted by V(u).
oi
LEMMA 4.8. Let c? (e) be summable on (0, 0), and let
(4.20)
(t) = hJe
t~0te
Let E be any constant. Then
(4.21) E C(Er) E(Err++7 )= 3 t)J3( () un if ? is an
even function,
(4.22) ETeJ.+r.'. ) = ) ,) ,t)ob if ? is an
odd function.
PROOF. For any suimable ? (z) we have, by (4.20)
t) t' (tt =a d ( bd el st)j. ,tt .
But the inner integral represents the density of the distribution
of U+ X,, Hence
Similarly
similarly
a00
J.
Hence
thiss follow (4.21) and (4.22) immediately because M (t) is
even or odd according as f (z) is even or odd. q.e.d.
LEMMA 4.9. Let f(z) and j (t) be as in Lemma 4.8. Let
j)
Hii
J(t) a'\St)g L:C CE ~)
frf)}^;I
Let f(t) 0(t"1) for large t. Then
if is even,
(4. 34) E Z7? I)JE y)ET(E7 A ,, ) ([+
if C is odd.
PROOF. From (4.231) and (4.22) we get immediately
(4.25) T{e w) E T' lP t ,\ (1.) ts) =. wojlft)(^
if is even,
(4.28) CE_ (J f.cgrt) Vt'r v.+ if is odd.
C4
By (4.f) and (4.X6) the integrands in (4.25) and (4.26) tend to
limits as n oa If (t) = 0(01), then J (t)g3( E t) = 0(t4).
Hence, by (4.1k) and (4.14), these integrands are dominated by
summable functions. It follows that the integrals tend to limits
equal to the integrals of the respective limiting integrands. q.e.d.
LFM A 4.10. For x ), 3 we have
(4.37) ()) = ,
(4.38) b 'xt) f() a
(4.29)
(4.30) tq
'e^
PROOF. (4.27) is true because its left side is equal to
P( ( SLT a x) while .EU J. Differentiating (4.27) twice
with respect to x we get (4.28). Write (4.27) in the follow
ing form:
Differentiation with respect to x once and twice yield (4.29)
and (4.30) resoeotively, q.e.d.
Ie are now in the position to prove the necessity of the
conditions (I), (II) and (III). Consider the following narti
oular functions V(z)s
cef1 %{ = I
f(
( x )z
(o rrespondingly
and, by (4.28) and (4.30),
TI
'e take C (z) to be P(1,jx) and z(C;x) successively in (4.23),
and take T (z) to be i( x) in (4.24). Then we get the follow
ing three equations, valid for x > 3 t
/  j !' L
J~L ^
T3^\e
21 ^ ^<
(4.33) (Uthjy/L n(Taut) EHA + 4f
There is no restriction in assuming G( ) 0. For convenience
77e assume that G(a6 ) > 0, but it is easily seen that the results
hold equally true for G( ) 0. Let
Let Z, V and I be the R. V's independent of U and whose T). F.'s
are respectively H(4;Z), HI(;q), and G(y)/G(ao). Then ,,e may
rite (4.31), (4.32), (4.33) as follows;
(4.34) p T t 7 ) =/ ))) ) Vj
. (4.35)) x)
'oe
A) (IF CY t v .4
/ \
(4.38) E c(EUt = 7 + c 3;x)}^)^
A 3P 0
o@
+1 73'? cft^ (7 4p H
= M t Cr 0
% '3 * ?
using (4.81), (4.22)
(with XI
replaced by Z, V or as3 the
oase arises).
For any R. V. X we have
Using this on (4.34) we get, for 6S,
IA3 P + U. ) i "C) Zr r >= /
14 qj
f'xt3%
e> PL Jf'v(Lx6L
In P 0 */' > a
9 7*P 1) +
tf both x and x are continuity points of G(y), we conclude
from the above, on letting E 0, that
In
f d4 Fn
ly '^
 2L 1Gw.
EIr
4 L Tr 2 q r ),jc
t^ 3yl, 0^
For any R. V. X with D. F. Q(y) and independent of U, it is
true that for 3e4 x,
SEzu&IA/ V) J( u )Q d V(y).
1EW 40 V, wt I <.x
(tmwl2
u 
X0^ J&L z+Jr
E1' "i c ) 
T31 .X^) CEc
S'
I1 I
I.) + x W d2') 9
c.3 )
I~I0(
Applying the above to (4.35) and (4.36) we have
(4.3 A {C f^ ^ 2 hJ~i) + Y6(tts
X 3 f j I.L c
"'^J Abi t/<.^
2 A ix
o jl.x I x3Jr c^isxt
') 4 4Htvh) d H' +,# x7 .. _.H ;
lt s d )t ) 19 ') .
;Sl
U T )E lt x)
A^^il0 1i
= I dfM) d Q( )jdo>
e^x y?^ Jff iz(t3
If +_ x, + (x + 3), t (x U 32) are continuity points of G(x),
then,on using (4.38), (4,39) and (4.40) are reduced to the
following:
y ()dc1z \f4y J^ d i*t) dt9&W), T
A E d fif ++f 6 )
Sjx i\^x Jfi &jc
c + pr, t) + d c, t o
oC l /( /xx 2 c, /l .txc3 
7 f dK@) j d t o(xt&t) I;ed4 P)
Hence
S .2,\ t
12 9% iXfd
Since the terms involving e tend to zero with S we obtain
the following:
(4.41)
& (Y~)de~)
2
a
(4.43) :Tr) 4r j Pd
We shall now deduce (I), (II) and (III) respectively from
(4.38), (4.41) and (4.42), E being chosen, Lemma 4.5 asserts
that there is N(E) so that Maxi i < s for n ( z H).
Then
If +x, +(x + ) are
(4.38) we have
all continuity
A n (x)= A)w +0
i 'i..
Z d[F1Cy)+e fc AFf,(LI).
points of G(y), then by
(A (tE) A x E)
whence, letting a o,
which proves (I). Again
>'
(4.43) Z
3
Sz J X )*  (JFnj( t
71<.X 2
where
d,
(1) 2
z,< o
k~A4.IjIc)c
 Uo )/""
'd7)\
{ A44
16rf
'E s(Ili x++
0F 6f(AFJ i
"e9'
* l^ JPI
i dFn I
4.38) we get
lyl /v
& E lC.2x+S)
7V ,
qu
5 E C2 eJ/) x).
Henoe, letting E * 0,
( I)
(4.41), (4.43) and (4.44) give
which is (II).
 Finally
CI)1
*.1
+
('J'x
(( /
6I~PJ1/
ci
1 10
'It
Using (
f" fx
 _x^i)
(4.44)
Cz)
 '1 ? :
<~~~ (7()(W, 2i[,)
Hene, by (4.d 42.)
The proof of the necessity part of the theorem is now
complete.
54 Some special oases. We shall consider a few interesting
oases in which A(x), B(x) and C(x) can be explicitly computed.
A(x) o, B(x) = 1, O() 0..
This is the central limit theorem, for which f(t) w .
(B) G(x) a 0, m = 0, Then
A(Lettng) = (x) = ge, ) .
This is the weak law of large nessity pabers, for whe th f(t) eorem is now
(A) G(x) = 0 for x < 1 and a for x > m Thea,
4 where a > 0. Then
A(x) = B(x) 0(x) 
[ ~ ~ ~ ~ ~ A d > a ^^
This is the Poisson case, where f(t) = ea(eit
(D) G(x) _e AV for x > 0 and
x < 0, 0 m = e where A > 0.
e
A(x) mAJ j9 B(x) AA0vc;e /
G(x) = 0 for
Then
o(x) W e
In this case
f(t)= F[c )z Ae
(E). o(x) r Io r
o Itt C\. Z2T ,C
A~~~ ^"'cor^^
() 
o
J~A~
fr.~
Cccm Cjc~ ?~
This is the case where f(t) = exp( co c, '/ ) /f/ , the
corresponding F(x) being the stable law discovered by Paul
Levy [L8].
o< jct G ^ ,.=c,r ^I ).
o .. CO [ Yco C. I
REFERENCES
1. W. Feller, "tUber den zentralen Grenzwertaatz der
Wahrsoheinlichkeitsyeohnung,"Math;. Zeit., vol. 40 (1935)
pp. 521559.
2. V. Glirenko, "Sul teorema limited della teoria delle
funjioni caratteristisohe," Giornale d. istituto italiano d.
attuari, vol. 7 (1936), pp. 160167.
3. B. Gnedenko, "Uber die Konvergenz der Verteilungsgesetze
von Swnmen voneinander unabhangiger Summanden," Comptes Rendus
U. R. S. S., vol. 18 (1938), pp. 231234.
4. A. Khintohin, "Zur Theorie der unbesohrankt teilbaren
Verteilungagesetze," Reoueil Math., 2(44) (1937), pp. 79119.
5. P. Levy, Calcul des Probabilites, Paris, 1925.
8. Theorie de l'Addition des Variables Aleatoires,
Paris, 1937.
7. J. Maroinkiewioz, "Quelques Theoremes de la theorie
des probabilites," Travaux de la Societe des Sciences et des
Lettres de Wilno, Olasse des Sciences llathematique et Naturelles,
vol. 13 (1939), pp. 113.
8. "Sur les fonotions independantes I, ?und.
Math., vol. 30 (1938), pp. 203214.
9. "Sur les fonotions independantes II, Pmd.
ath., vol 30 (1938), pp. 349364.
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rlg:V *iff NmmNsmD M t nwffMm11mWME2
$4i W l MiinT:
1. F (x) PaWffif (:Pj({V i x FiR F( ) 0,
F(co) = 1).
2. ,(,) A F(x) ,9' Fe i (ap P(t)=, f d F(x)).
oo
5. M(, (F') ^F (x) t # P&'Sm (Sp Mi (F) = x d F (x) .
4. Ly (F) I  ix log 1 d F(x).
5. Hn(t) MW3f6SK ].
6. (f)= ( 1)' ()f ((I < ) t).
7. nJ /f(t) A a? ^flJ Pk(t) f T:
Pka (t) = t rf (0).
oo
A=: M n / O,p 1 > 0, Xinj[iJ 2n > 0, :AIAiJ F(x) t 0i
()n.(F f1 2nf+P1 d y fe H2, (t)f (y t) d t.
0r 0
Wa 2.1 (g.gIE,:+Vi iJ) M % (F) R.(k j@&( 0,
o < & < 1), m= A t0 ff Cr^ f (t) FMAflftam^:
f(t)=1+alt+ ... +aktk+o0 t k+yp (t)>.
:&R, *.nAIMfTAIm7T, a;
[.]+ k k ]+
k+y(F) 1I'(k+,y+1)sin ( [2 2)
kl+y+l t
0
_.I 2.2 (!e 2.1 I^' iJ) tng t=MO f( F)m~ ]f (t) Ti MFm :
f (t) = 1 + a t + +aktk + 0 t(] tk+ "), (a > 0),
95 ,I+y(F J rP P q fp a M. ENE.
g 5.1 (3.E132 All) M2 (F) ifR (n ZiE& ) 4
fT,_ (f)=o (t2).
2 5.2 (5N 5.1 89I919J) R M!n (F) fPSR (n (_EI ), .ZTiffaUf (t)
f (t) = 1 + a t + + a2ni t2n1 + 0 (t2n).
2a 5.5 (5f 5.2 lZ'Tl) k M2n(F) ASR (n E ) iJla,
f (t) AFTiMIR:
f(t)3= 1 + ait + (F. + an t2n + o (t=n), (t 0).
5SA 5.4 A M27 (F) $; PR (n .). (n t=0 MW9 l g I f (t)} Fi
O9MA,
P. L. HSU
Vol. 1
No. 5 ABSOLUTE MOMENTS AND CHARACTERISTIC FUNCTION 259
q (.f (t) } = q { P2n (t) I + 0 tn p (t) .
Rm 5.5 '~A L2, (F) YR (n E1 0) 0 1,,Rffl t=o0 WXRWI f(t)
f(t) = 1 + al t + .* + a2. t + 0 (tn ()).
*iSM! 4.1 (i'linii i0 JEij .Tl'1.i ) 'k Mn+i(F) SjJl" (n 1E !( R
0) ,mlRlfEi t=0 E.ir 8 ( ft)} Fl'i'9Ef1]t
(f(t) 1 = 1 + a t2 + a t4 + + an t2n + 0 \ t 2n+1 j) (.
*Tle, AAn'^sflim^, W
M1,+i (F) = ( 1)"n+ 2 1 (2n + 1) !' f(t)2n+ (t) d t.
o
0
W 4.2 AF L2n+i (F) 3TW (n ~1.c 0) m", t=0 ff M f(t) T
f (t) = Pn+1 (t) + ] t ,n+ t (t)}.
r5, Z n, t=0 mfAN f~t) W Tff IRN :
f (t) = I + al t2 + a2 + t2 + + a2n+l te"+ + 0 1 t 12n+l (t) W,
XAnT'' x < 0 flp F(x)=0, N, L2n+ (F) < oo.
5a 5.1 0n 1 !f(t) } M 0, gff Mi (F)=oo.
ABSOLUTE MOMENTS AND CHARACTERISTIC
FUNCTION.
BY P. L. Hsu
Peking University
1. Introduction and definitions. In this paper we study the relation
between the absolute moments and the characteristic function (c.f.) of a
distribution law. A distribution law (which for brevity will be called "law"
throughout this paper) is a function F(x) defined and nondecreasing on
the whole line co < x < co and satisfying the condition
(1) F(co) = 0, F(oo) 1.
The c. f. of a law F (x) is the function
(2) f(t) =. e= d F (x),
oo
and the absolute moment of order f (f F 0) of a law F(x) is the quantity
(5) M( (.F)= x !fPdF(x).1)
We shall also consider the quantity
(4) L (F) = f x P log x I d F (x),')
lxil>e
but only for nonnegative integral values of j.
The evident facts that f ( t) =f(t), If(t) \I 5 1 =f(0), that 9T f(t) }
and g If(t) are respectively even and odd functions, will be used
frequently in this paper without any explicit mention. For a fuller
knowledge of laws and cf.s we refer to the books by P. Levy [4] and H.
Cramer [1].
In the following investigation we shall establish various identity
relations between M, (F) andf(t), and various necessary and sufficient
conditions, in terms of the c.f., for the finiteness of the absolute moment
1) Mo(F) is defined as F(oo)F(oo)=1. If ,8<0 and F(x) has a discontinuity at
x=0, then Mp(F) is defined to be oo. If < 0 and F(x) is continuous at x=0, then in the
definition (5) of M! (F) the point x=0 is omitted from the domain of integration.
2) If fl > 0, we may as well consider j\x log [x d F (x) so far as the finiteness of
the latter is concerned. The precaution 'Fl >e is made only to avoid the anomaly at xr0,
when #0.
P. L. HSU
Vol. 1
ABSOLUTE MOMENTS AND CHARACTERISTIC FUNCTION
of a given positive order (including one for the finiteneis of L, (F)).
Some results about moments of negative order are also touched upon.
If M1 (F) < oo, where k is a nonnegative integer, then, as is immedi
ately seen from the permissibility of differentiation under the integration
sign in (2), f(t) is ktime differentiable, and
1 1
(5) f (t) = 1 + tf' (0) + 2 t2f" (0)+ ... + ... t f( (0) + (tk), .(t 0),
with
(6) f(v)(0) i" xdF(x), ( 1,2,...,).
Fortet [2] has shown that: if k is even and if f(t) admits an expansion of
the following form
(7) f(t) = 1 + alit + ... + ak (it)k + o (t), (t 0),
then Mk (F) < co if k is odd and if (7) is replaced by the stronger con
dition
(8) f (t)= 1 + a it+  + a (it)k + 0 ( [ t k+), (e > 0),
then M, (F) < co. Although several of our theorems include Fortet's
theorem as a special case, their proofs are much simpler.
At the end of this paper we shall establish a relation between the
Stieltjes transform and the cf.
2. Absolute moments of positive fractional order. We shall briefly
indicate the derivation of the following known formula:
(9) e_'& On
where Hn (t) is the Hermite polynomial:
(10) H, (t) ( et2 L e'.
Starting from the identity
No. 5
P. L. HSU
(11) eitx? e J: dx 2 1/"C et
00
which may be verified by expanding ix and integrating term by term,
we differentiate it n times and obtain
(12) ( i)n x" eii dx = ( 1)n 2 / Hn () e'.
oo
Formula (9) is then derived from (12) by the Fourier reciprocity formula.
In (9), replace x by xy and n by 2n:
(135) xn ey22 ( 1)ny2" fetyt2Hn(t)dt.
Let F(x) be a law; integrate both sides of (15) with respect to
F(x):
(14) fx2n e 2x2 ( d F (x) f eiy H (d) ti
_( )ny2 e,t2 H (t) dt eifYzdF(x)
yn e,2 H", (t) f(y t) d t,
the change of order of integration involved in the above computation being
easily justifiable.
Let {? be any positive number, multiply both sides of (14) by y,i'
and integrate with respect to y over 0 < y < oo : If either 2n  > 0
or F(x) is continuous at x= 0, then,
Vol. 1
ABSOLUTE MOMENTS AND CHERACTERTSTIC FUNCTION
fyP1d y J xneYzdFF f y dy x2neVY2 d F(x)
0 0o 0 1z10
= fn d F (x) fyf1 ei22 d y = f x 2j P d F (x)f uPl e1 du
Irl>0 0 isl>0 0
2P1 r (7(F),
which is to be equated to the result of the same operation applied to the
right side of (14). Therefore, if P > 0, n is nonnegative integer, and
if either 2n9 > 0 or F (x) is continuous at x = 0, two have
(15) Mn_ (F) l)Tz21P Y +l dy e 12 (t) f (y t) d t.
By means of (15) the absolute moment of any order is expressed in terms
of the ef.
Let k+y (k_ 0 and integral, 0 < y < 1) be any positive fraction.
Setting n= [] + 1 and / =2[] + 2ky in (15) we get
(16) Mk+F) ( 
X yk)dy j e,2H2 +2(t) (yt) dt.
f I) f
Both formula (16) and the following Lemma are preparatory for
the proof of Theorem 2.1.
If a c.f. f(t) is ktime differentiable, we shall always denote by P. (t)
the following polynomial:
No. 5
S1
(17) Pk (t) = t fXV) (0).
LEMMA. Let F (x) be a law, f(t) be its cf., k be a nonnegative
integer, and 0 < 7 < 1. If Mk (F) < co, then
(18) f(t) Pk (t) + O ( t +).
Proof. The finiteness of Mk+r (F) implies that of M, (F) and hence
the ktime differentiability off(t). We 1have
fP() (t) (i x)7 ei x d d F (r),
The above inequality proves the Lemma in case k= 0. For A > 0 we
have
If (t) PA (t) (0) ( ) a (e 1) ( F (0) ,
S (C ) (It ()k( 1 ) (k) ) fi(k)t(0) F(d ) 2 ( sin u 1 C t (k+y)
0 0
3 From now on we shall let the letter C denote any positive constant, not necessarily
the sabove at each occurrence.
the same at each occurrence.
P. L. HSU
Vnl. 1
ABSOLUTE MOMENTS AND CHARACTERISTIC FUNCTION
Hence the Lemma is proved.
THEOREM 2.1. Let F (x), f(t), k, 7 have the same meaning as in the
preceding Lemma. In order that Mk+. (F) < co, it is necessary and
sufficient that, in a neighborhood I t j < 6,f (t) should admit an expansion
of the following form:
(19) (t) = Qk (t) + O( t + (t)),
where Qk (t) is a polynomial of degree k, and ip (t) is a function defined,
nonnegative and bounded for 0 < I t < (5 and satisfying the condition')
(20) f t 11 ilh (t) d t < co.
s
Further, if (19) and (20) are fulfilled, then Q4 (t)= P (t) and
(21) MAk+(F)=( 1)L+12 F(k+y+1)sin (k [+] )
j tk+r+1
0
Proof The condition is necessary.We need only show that,
assuming M ., (F) < oo, we have
(22) f t 1k11 f (t) Pk (t) dt < co,
00~
for then the functions
Qk (t)= Pk t) = ik If(t) PA (t)
will satisfy (19) and (20) and, besides, ip (t) is bounded, by the Lemma
just proved. To establish (22), set
4) Examples of Vy (t) are (log) log  ( log log etc., with c > 0.
No. 5
k i. Vl
A1
(23'5) E, (t) = (i t)v.5)
Then
f(k+r(t) d t i k++ et Ek (tx) I d F(x)
_< t+ ]fetxE7,(tx)dF(x)fdF(x) E(tx) dt
= k+YdP(x) jf I (u 1< 00,
which proves (22).
The condition is sufficient.The equations (19) and (20) together
imply that
t *71 if (t) QA(t) I dt < o.
Then,
(24) f t 1ky1 /(t) Qk(t) dt < o,
because the part of the intetal extended ever ItI > 6 is evidently
finite.
By virtue of the fact that
(25) fe'_t2 Hm(t)ttdt=O for 0O 1
(which results from the orthogonality property of the Hermite poly
nomials), we may write (16) as
5) We shall stick to this notation throughout the rest of this paper,
P. L. HSU
Vol. 1
ABSOLUTE MOMENTS AND CHARACTERISTIC FUNCTION
(26) M,;+ (F) = C fykyI dy e"H2 [L+2(t) f(yt) Qk(yt) dt.
0 00
Therefore,
Mk+r(F) C Y1dy fe2 I H2[ +2 (t) I f(yt) Qk (yt) Idt
0o 00
C f e' I H[A +2 (W dt fykr1 if(yt) Qk(yt) I dy
 0
By (24), M+ (F) < co. Thus the sufficiency of the condition is proved.
Then f(t) is ktime differentiable and so we must have Qk (t) =P4 (t).
Replacing Q, (y t) by Pk(y t) in (26) and reversing the order of integration
we obtain
(27) Mk+r(F) C fe,2H (dt [r 1 f~ yt)Pk~yt}\dy
(27) M+(F)= C fetH] ]+2(t)dt fy 1 f(yt) Pk(yt) I dy
00 0
= CJ et2 H[]2 (t) d t yy19 if(y t) Pk, (y t) dy
o o
0 0
=C e H2[.+2 (t)tk+ d t ukr1 l f(u) PA (u) du
0 0
= AJ u1 f f(u) P, (u) d 4u,
where A is some constant independent of F (x). We choose the particular
pair
No.
P. L. HSU
1 1 1
F(x) = Jel dy, f(t) = etI dx = 1 +
For this pair we have
Mk+y(F) = I x 1 ik+yell dx = r(k + y + 1),
00a
f (t) P (t) = ( 1)[ ]+ (1 + t2)1 t2[]+2.
Substituting in (27) we obtain
(28) F(k + y + 1) = A ( 1) l+1 t2[]k+1r (1 + t2)ldt
0
(1)[ ]+1aA
sin( k_[ +
(27) and (28) together lead to (21). The proof of Theorem 2.1 is thus
complete.
We note the following special case of Theorem 2.1:
THEOREM 2.2. If in a neighborhood It 1 < 6 we have
(29) f(t)= Qk(t) + O(] t Ik+), (a > 0),
where Qk (t) is a polynomial of degree k, then M,+y (F) < co for every
0 < y
This theorem, which follows from Theorem 2.1 on putting
'ip (t)= It l"a', contains Fortet's result, referred to in 1, that (29) implies
the finiteness of M, (F).
5. Moments of nonnegative even order. If 0 (t) is any function
defined for co < t < co, we write
(30) n)t)
vi2
Vol. I
ABSOLUTE MOMENTS AND CHARACTERISTIC FUNCTION
The quantity ["3 (0) = lim t" T, (0), if it exists, is known as the generalized
to
n th derivative of (t) at t = 0. In this connexion two familiar facts are
that (0) = (n,(0) provided the latter exists, and that T, (4)=0 for an
even or an odd function 9 (t), according as n is odd or even.
When we apply T to e*, regarding x as constant, we get
(31) T2e)  ((e = eit)= (2 i)n sinn tx.
We also have
(52) T.(tz)=0 for 0 l
To see this, observe that the right side of (51) is a power series beginning
with the term tn. If we apply T, term by term to the exponential
series eil", the resulting series must also begin with the term tin, and
this implies (52).
THEOREM 5.1.) In order that M,, (F) be finite, it is necessary and
sufficient that we should have
(5533) T2n (f) = O (t""),
where f=f(t) is the c.f. of the law F(x).
Proof. The necessity of the condition follows from the existence of
fi2n (0) =f(2) (0). To prove the sufficiency, we have, by (51),
If now t2n" TI (7 )  K, then
(55) K > J (sinFtx ( dF(x) > f ( sln )" xndF(x).
But jI 1 >  for 101e < ". Applying this to (55) we get
6) Compare with Levy [4J, p. 174.
No. 5
(56) K (> f xddF (x).
Ix
Making I t  0 in (56) we conclude that M,, (F) is finite.
The following two theorems are simple corollaries to Theorem
5.1.
THEOREM 5.2. A necessary and sufficient condition for the finiteness
of MK, (F) is that
(57) f (t) = Q2n1 (t) + 0 (tn),
where Q,i (t) is a polynomial of degree 2 n 1.
For we have, by (52), T,. (Q__,_)= 0, whence T2. (f) = O (t").
THEOREM 5.5. A necessary and sufficient condition for the finiteness
of M, (F) is that
(38) f(t) = Q(,,(t) + o (t), (t 0),
where Q, (t) is a polynomial of degree 2 n.
This is Fortet's criterion for the finiteness of an evenordered moment,
as referred to in 1.
THEOREM 5.4. In order that M2, (F) < co, (n > 0), it is necessary
that, in a neighborhood It j < 6,
(59) 9 {f (t) } 9 {P2(t) I + O(t2p (t)),
where tp (t) satisfies the same conditions as in Theorem 2.1.
Proof. We need only verify that
fI t 1n'I I{f(t) P2n(t) } I dt =
Vol. i
P. L. HSU
ABSOLUTE MOMENTS AND CHIERACTERIIS'1C FUNCTI'ION
= \t 2n1 IfW (t) n (t) dt < co.
Now,
t 2n f(t) 2
f I 't nldt f ( 9 eW Eni ( tx\) \ d F(x)
f_ I t ]2n1dt f eitz E2n1(t)d F(x)
= x F(x) f !2n1 e E (u) du < ,
which gives proof.
Theorem 2.1, which gives a criterion for the finiteness of a fractional
ordered absolute moment M+,, (F), is no longer true for integralordered
absolute moments (y7=0). The counter part of Theorem 2.1 for the case
k=2 n, y= 0 is the following
THEOREM 5.5. In order that Ln (F)7 be finite, it is necessary and
sufficient that, in a neighborhood It I < 6, we should have
(40) f(t) = Q,2 (t) + 0 (tn p (t)),
where Q2,,(t) i a polynomial of degree 2n and p (t) satisfies the same
conditions as in Theorem 2.1.
7) For definition see equation (4).
No. 5
Proof. The condition is necessary.Since the finiteness of L,, (F)
implies that of M,,, (F), we may set Q, (t)= P2 (t). We need only verify that
1 1
(41) t n If(t) Ps(t) d t = 2 t2 f(t) P2n (t) I t < co
1 0
We have
1 1
tn1 f(t) p2(t) dt t2dt eiz E2(tx) d F(x)
0 o 0
1 o
S'tJnld t eIit r E2n (tx) d F (x)
0 
= d F (x) t2n1 eit E2n (tx) d t
0 0
== x2 d F (x) u_2z11 i" E ,(u) du
zlXe
C + Ln fd F (f) ja u2n1 I e'" E92 (it) I d u
lxs>e
< C + Jx2n d F (x) C I1 du=C + C IJ.xlog 1 x I d F (.r) < co,
ix >c e lxl>e
which constitutes proof.
The condition is sufficient.By virtue of Theorem 5.2, the condition
(40) implies the finiteness of MA/, (F). Therefore Q,, (t) = P2,, (t) and we
have
1
ft'2 I\f (t) .,. (t) dt < co.
0
P. L. IISU
Vol. 1
ABSOLUTE MOMENTS AND CHERACTEIilSTIC FUA'CTIONV
whence, a fortiori
1
(41)a ''1 ] f (t) PJ,, (t) } 'dt < co.
0
Since l eu E, (t)j is a function which does not change sign, we
have by (41),,
co > tn1d t eilx E2. (t x) d F (x)
1 o
= t2n1 d t f le{e E2n (t) ) dF (x)
0 o
0 1
= d dF(x) t21 e E2(tx) .
0 0
= J x2" dF(x)J u2n1  ei" E2n(u) \du
00 0
^ J'."2 F ( 2"1 )i ei" F'_^ (a,) 1 d u
lj ,:> <' e
> fx2ndF(x) C C 1 du = C C x2z log x dF(x),
Ixl>c e
which implies the finiteness of L,7, (F).
4. Absolute moments of positive odd order. We login by giving
an example showing that. the (2 n + 1)time differentiability of the cf. at
t = 0 does not imply the finiteness of M!_+, (F).
Let
X'( = (i* ') ) = 0 ([ lg < ),I
No. 5
274 P. L. IHso Vol. i
fx
F(x) =c p (y) dy,
where c is a constant so determined as to make F (co) = 1. Then
f (t) 2c cost dx, f(2n) (t)= : 2c x dx,
f(t) = c +2 lo+ x J xi2logx
e t
f() (t) fn) (0) 2c f cos 4 c si 2
+c f d f dr t Etdx lf dx
t t x2 ogx1t log x t logx
e e
4c I' dx dx 4c t dx
\t aog x f logix logtl f log x
K(1 e e
2c 4c
log tl log I t log t I
Therefore f(n+') (0) 0 ; but plainly M"'+ (F)= 0o.
THEOREM 4.1. In order that M,2+, (F) be finite, it is necessary and
sufficient that, in a neighborhood ItI < 6, we should have
(42) f(t)} = Qn(t) + 0 t J2n+1,p (t) },
where Qn (t) is a polynomial of degree n and tp (t) satisfies the same
conditions as in Theorem 2.1.
If (42) is satisfied, then Q, (t2) [ {P,, (t)\ and MiWA (F) is given by
(21) with k=2n+ 1, y=0:
(45) M2,+1 (F) == ( 1)n+1'2~r1 (2n + 1) Ift2 f(t) P2n (t) d t.
0
Proof. The condition is necessary.If WM,., (F) < co, then we set
Q.(t)=9 P2,,(t)j and verify that
ABSOLUTE MOMENTS AND CHARACTERISTIC FUNCTION
tJn2 J 9 ife(t) P2n(t) I dt
= Ma+1 (F) u2n2 { ei, E2n(u }) du < oo.
The condition is sufficient.Since % {f(t)] is the c.f. of the law
G (x)= {F (x) + 1 .F( x), it follows from (42) and Theorem 5.2 that
M%, (G) =Mn (F) < co. Therefore Qn (t2) =91 {Pt (t)}, and our condition
asserts that
J tf\ if(t)P2n(t)}ldt= f+ < Co.
Itl< J Itl>J
Since {etE,,(t)j is a function which keeps a constant sign, routine
computation leads to
i f(t)~P2n+.) dt= M2n+lt1 (F) I E22n+(u) I d
St2n2 u2n+2
wherefrom the finiteness of M2o+1 (F) follows. This proves the sufficiency
of the condition (42).
A similar computation gives
00 00
ft2n2 If f(t) P (t) dt=M n+1 (F) fu2n2 R eiu E2n (u) d u,
whence
Mn+ (F) = A1f t2n2 \f(t) P2n(t) } dt,
W
No. 5
where A is independent of F(x). The device we used to determine the
constant 4 in (27) can be used to determine the constant A1, and it leads
to (45). This completes the proof.
The analogy to Theorem 2.1 for the case k = 2 n + 1 and y =0 is the
following.
THEOREM 4.2. In order that L2nl (F),' < co, it is necessary that, in
a neighborhood It < 6, we should have
(44) f(t) = P2+1 (t) + O t 12+1 (t) },
where Vp (t) satisfies the same conditions as in Theorem 2.1.
Conversely, if in a neighborhood 1 tI < 6 we have
(45) f (t) = Q2n+1 (t) + 0 + I t jin+1 p (t) I,
where Q2,,,, (t) is a polynomial of degree 2 n + 1 and Vp (t) satisfies the
same conditions as in Theorem 2.1 and if further F (x) = 0 for x < 0,
then L20+, (F) < co.
Proof. To prove first part of the theorem, we need only verify
the relation
lf(t) P2 +1 (t) d t < co
0
under the assumption that Lg,,,+ (F) < co (which implies the finiteness of
1,,, (F) and hence the (2n + 1)time differentiablity of f(t)).
We have
ft2n2 If(t) P2,+1 (t) d t
< f I 12n+2 d F(x)J U2n"2 "' En, +1 (1u) ] d u
8) For definition see equation (4),
P. L. HSU
Vol. 1
No. 5 ABSOLUTE MOMENTS AND CHARACTERISTIC FUNCTION 277
[xl
S2 1 .r 2'1 d Fx) U2112 1 el" ,+ () d u
SC + fx :' l1 dF(x) Cu du=C+ CLn+i (F) < o,
which gives the proof.
To prove the second part, observe that, by virtue of Theorem 4.1, the
condition (45) implies the finiteness of l.+ (F). Therefore Q,,, (t)=
P 1, ,, (t) and our condition asserts that
1
ft2n2 f (t) P2, 1 (t) d t < co,
0
whence, taking into account the condition that F(x)= 0 for x < 0,
(46) f t2"2Odt J eitx E2n+1 (tx) dF(x)
0 u
1
=ft'21I q f (t) P++1 (t) dt < co.
0
But, when x 0 and t 0, the function g felE ,, (tx)] keeps a
constant sign. Therefore, by (46),
o00 > t"2 dt J { et E2.+ (t) ) d F (x)
o o
0 0
1 t" OF(x) e  2+I(t (u)  du.
0 0
P. L. HSU
x x2"+~1 d F(x) fu'2 I g Se" E2n+1 (u) d i
e e
S x2"n+l dF(x) C uldu C = C Ln+1 (F) C,
e e
which implies the finiteness of L2n+1 (F).
5. Absolute moments of negative order. If F (x) is continuous at
x 0, then, putting n =0 in (15) we obtain
(47) M_(F) 21 y' dy fe'f(yt)dt
Y rr(4) 6o _
a formula expressing a general absolute moment of negative order in
terms of the c.f. We shall not investigate further along this line, except
note the following readily proved but somewhat curious theorem.
THEOREM 5.1. If f(t) has a nonnegative real part, then Ml_ { (F)
For if 9 \f(t)} 0, then we have
M_1(F) = e Je dtJ If(yt) Idy
0 0
= ft etdt fJ ff(t)dt= I o.
0 0
6. The" Stieltjes transform. If F(x) is a law, the function
(48) L(t)= fd(, ( o < t < oo)
x +it
Vol, 1
No. 5 ABSOLUTE MOMENTS AND CHERACTERISTIC FUNCTION 279
is known as the Stieljes transform of F (x). We have
(49) L () dF(x) dF(x) e(' )sds
o00 00 0
= es s eitsx d F (x) = es ds ez d F () = fet d F (x),
0 0 0 0 0
where
(50) F1 (x) e= F eF ds
0
is again a law. Thus L (~) (which is defined to be 1 at t=0) is the
cf. of F1 (x). Besides, for j > 0 we have
(51) Mp (Fi)= fesds f xPdF (d )
0 00
=f sPe fds f\xFdF.(x)= r(+ 1)iM(F).
0 c
It follows that My (F) is finite if and only if MA (F,) is so. Accordingly,
corresponding to every theorem (except the two theorems about Lk (F))
in 2,5,4 there is a theorem in which L (t) plays the part of f(t). e. g.
the theorem corresponding to Theorem 2.1 reads:
In order that M+, (F) < co (k > 0 and integral, 0 < y < 1), it is
necessary and sufficient that for It I > K we should have
(52) L(t)= . + C + + + Cp + 0 t Ir(t)S,
where 7 (t) is defined, nonnegative and bounded for I t > K and satisfies
the condition
(55) f dt < .
It.>K
If (52) and (55) are satisfied, then
(54) M F)= ( ) 2 1 sin [k + )
x tky L (t) ( CA (t dt.
The two theorems corresponding to Theorem 5.4 and Theorem 2.2
(taking the special case y=0) constitute Hamburger's Lemma on the
Stieltjes transform.')
REFERENCES
[1] H. Cram6r, "Random Variables and Probability Distributions" (Camb. Univ. Press 1957)
[2] R. Fortet, Calcul des Moments d'une Ponction de Repartition a Partir de la Charact6risti
que. Bull. des Sci. Math. (2) (1944), vol. 68, pp. 117151.
[5] H. Hamburger, Uiber eine Erweiterung des Stieltjesschen Momentenproblems. Math.
Ann. (1920) vol. 81, pp. 255519.
[4] P. Levy, "Calcul des Probabilitds." (Paris 1925)
(RECEIVED JANUARY 19, 1951)
9) Cf. Hamburger [5], p. 267, or Fortet, loc. cit, p. 118.
P. L. HSU
Vol. 1
7
1. 51 AE~nMM4U@ (J F ) W
S4 f 1 M 1954 ) 3 J
A T(E) Vlt (m), fmt) (m^ .
(Marcinkiewicz) ,ka ^W .(T^^W) F)
SerxdF < 00 ( f erxdF < co) (1)
J q\ 0I! )
' y ^%1ER, MI^M%( (U).)23)
"r^,WWW^h'^ ^'itW^%, @a "*PM (U)" tw'q "M4
@$RA@ 2, 3, 4 HOi,^M^HM (0) ]j <(iJ. E
1953 4 I 24 E]$Ct1.
W E [2], [3] 170 0, AE I [11 22 W /
2) O [4] E1 1 3, MijiiaMR 3.
3) [11] 21W. /
(In x)", In x (In in x)A, *, (A > 1) (2)
&f 55 a, f 9 FM Pi M N M W N 4k A
2. F m 1. 1 g() 4ZA t m9 figW1^jE t %.Pbun7V,4
,$ ,f.g g(o) = 1. tJ g(0) Jlf@i.
AM4 .4 RIK g(t) m4 i 'U(i g(t) = 1) ir
g1 (t)= g() 'g(oo)
1g(oo)
A A af (g (o) =o), g4@I g(t)=g(o) +
(1g(oo)) gi(t) A{@^.
5 a 2. L^a 1 ^it@(igfM 1)i (U).
*a g(t) Ammi,Xis a> o0 g(t) 4M {,M
g*(t)=g(t), It[ a
p ( ) *(t) i g(s) (P g*(,) > g(t), Ill> a)
XSIi Iel < I
iltc i@Mit exp ( t I),
.
exp { (coici sgn t)1a, (o 0, \ciI < co tg2 j
co= 1 M WW
4) A* [5] 170 W.
