COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI, 21.
ANAL)-TIC FUNCTION METHODS IN PROBABILITY THEORY Edited by:
B. GYIRES
NORTH·HOLLAND PUBLISHING COMPANY AMSTERDAM - OXFORD-NEW YORK
PREFACE
This book comprises the proceedings containing detailed versions of most of the papers presented at the Colloquium on the Methods of Complex Analysis in the Theory of Probability and Statistics held in the Kossuth L. University of Debrecen, Hungary from Augustus 29 to September 2,
1977 as well as some others which were
submitted later. All papers in this book were refereed. The Organizing Committee consisted of B. Gyires (chairman), P.
Bartfai
(secretary), L. Tar (secretary),
M. Aratb, P. Medgyessy, P.
R~v~sz,
K. Tandori, J. Tomkb,
I. Vincze. There were 49 participants at
t~e
Colloquium from
10 different countries, including 19 from abroad. I wish to thank Professor E. Lukacs for his suggestions throughout the organization of the Colloquium.
Thank is also due to Dr.
P.
Bartfai for taking
charge of the correspondence.
B. Gyires
-
3 -
ClJNTENTS
PREFACE
3
COHTENTS
5
SCIENTIFIC PROGRAII ..
7 I I
LIST OF PARTICIPANTS
P.
Bartfai, Characterizations by sufficient statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H.
Bergstrom, Representation of infinitely divisible probability
~easures in
~2
and some of its
subspaces. . .. . . . . . . . .. . . .. . .. . . . .. . . .. . . . . .. .. E.
Csore8 -
L.
Stacho, A step toward an
43
~symptotic
expansion fo( the Cramer - von l!ises statistic
z.
Daroczy -
W.
Eberl, Recursively defined l1arkov processes
53
Gy. iMaksa, Nonneeative information
functions ..
(discrete T.
21
Csaki, On so~e distribution concerning maximum and minimum of a Wiener prOcEss..............
S.
15
67
para~eter).......................
79
Gerstenkorn, Distribution of the sum and the mean of mixed random variables in a class of distributions . . . . . . . . . . .
z.
Govindarajulu -
A.P.
93
Gore, Locally most powerful
and other tests for independence in multivariate populations........................ B.
Gyir~s,
99
On a generalization of Stirline's numbers
of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1L:3
B. Gyires, Constant reeression of quadratic statistics on the sample mean, W.K.
Hayman -
II . . . . . . . . . . . . . . . . . . .
I. Vincze, Uarkov-type inequalities
and entire functions . . . . . . . . . . . . . . . . . . . . . S.K. Katti,
137 153
Infinite divisibility of discrete
distributions,
III ..
165
5 -
S.K. Katti - J. Stith, An empirical graph for choosing a population distribution using a characterization property •..••.......••••.....
173
K. Lajko, A characterization of generalized normal and gamma distributions ....••.•.•.•...•.••....
199
E. LukAcs, On some properties of symmetric stable distributions . . . • . . . . . . . . • . . . . . • . . . . • . . . • . • . . . 227 J. Panaretos, A characterization of a general class of multivariate discrete distributions •..•.... 243 J. Panaretos - E. Xekalaki, A characteristic property of certain discrete distributions .... 253 Gy. Pap, On the asymptotic behaviour of the generalized binomial distributions •.••.•.....•...... 269 B. Ramachandran, On the strong Harkov property of the exponential laws ....•.....••••.•. , . . . . . . . . 277 B. Ramachandran, On some fundamental lemmas of Linnik •••••.....••.•.....••...••.•.•.......... 293 V.K. Rohatgi, Asymptotic expansions in a local limit theorem ••••........•.•..•..•....•..•••......•. 307 K. Sarkadi, Characterization and testing for normali ty •••••••..•..•...•..........•......... 317 V. Seshadri - G.P.H. Styan, Canonical correlations, rank additivity and characterization of multivariate norm§llity •••.••..•......•......•...... 331 F.W. Steutel, Infinite divisibility of mixtures of 345
gamma distribution S.J. Wolfe, Mixtures of infinitely divisible
distribution functions ..•....•...•..•....•.•.• 359 E. Xekalaki, On characterizing the bivariate Poisson, binomial and negative binomial ~
distributions •••••...••......••••.•..•....•.•. 369
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6 -
SCIENTIFIC PROGRAM
29. August 2.00 -
3.00 p.m. Opening
3.00 -
3.30 p.m. E. LukAcs: On some properties of symmetric stable distributions
3.30 -
4.00 p.m. H.
Bergstrom: Representations of
infinitely divisible measures in
Rk
and in the Hilbert space by Gaussian invariants 4.00 -
4.30 p.m. K.
Sarkadi: Characterization and
testing for normality 5.00 -
5.30 p m.
I. Vincze: On a probabilistic problem concerning
6.00 p.m.
5.30 -
e~ire
functions
S.J. Wolfe: ri1ixtures of infinitely divisible d~stribution functions
\
30. August Chairman: H. 9.00 -
9.30 a.m.
Bergstrom B. Gyires: Constant regression of quadratic statistics on linear statistics
9.30 -
10.00 a.m.
W. Eberl: Convergence of recursive defined Markov processes
10.00 -
10.30 a.m.
o.
GulyAs - G.
L~grAdy:
Sampling
theorems for homogeneous and isotropic random fields 11.00 -
I I .30 a.m.
B. Ramachandran: On the strong Markov property of the exponential laws
11.30 -
12.00 a.m. J. Panaretos! A characterization of a general class of multivariate discrete distributions
-
7 -
Chairman: 3.00 -
B.
Ramachandran
3.30 p.m.
S.K.
Katti: An empirical graph for
choosing a population using a characterization property 3.30 -
P.
4.00 p.m.
Bartfai: Characterizations by
sufficient statistics 4.00 -
F.W.
4.30 p.m.
Steutel: Mixtures of gamma
distributions Chairman: 5.00 -
I. Vincze D. Dugue:
5.30 p.m.
Characteristic functions
in analysis of variance and design of experiments 5.30 -
K.
6.00 p.m.
Lajk6: Char8cterizations of
generalized normal and gamma distributions 31. August Chairman: T. 9.00 -
9.30 a.m.
9.30 -
10.00 a.m.
Gerstenkorn V.K.
Rohatgi: Asymptotic expansions
in the central limit theorem S. Csorg8: On an asymptotic expansion for the Cramer von Mises statistic 10.00 -
10.30 a.m.
A.
Szep: Random power series with
weakly dependent coefficients Chairman: V.K. 11.00 -
11.30 a.m.
Rohatgi Gy.
Pap: On the asymptotic behaviour
of the generalized binomial-distributions
- 8 -
31.
August
11.30 -
12.00 a.m.
V.
Seshadri:
A theorem on cha-
racterizing the Chairman:
3.00 -
S.S.
law
Wolfe Z.
3.30 p.m.
~ormal
Daroczy:
Uber die Characterizierung
der Entropy
3.30 -
B.
4.00 p.m.
Forte:
Non-symmetric entropies and
random variables
4.00 -
E.
4.30 p.m.
Csaki:
On some distributions
concErning maximum and minimum of Wiener process Chairman:
5.00 -
V.
Seshadri
5.30 p.m.
V.M.
Zolotarev:
On representations
of mathematical expectations
5.30 -
6.00 p.m.
J.G.
Szekely:
(
On a Chernoff t)(pe
function
1.
September Excursion
2.
September Chairman:
V.M.
9.00 -
9.30 a.m.
9.30 -
10.00 a.m.
Zolotarev H.
Kac:
Some probabilistic aspects
of potential theory S.K.
Katti:
Infinite divisibility of
discrete distributions,
10.00 -
10.30 a.m.
E.
Xekalaki:
Part
III.
On characterizing the
bivariate Poisson binomial and negative binomial distributions
-
9 -
Chairman: E. 11.00 -
11.30 a.m.
Lukacs
B. Ramachandran: On some fundamental lemmas of Linnik
11.30 -
12.00 a.m.
T.
Gerstenhorn:
Distribution of the
sum and the mean of mixed random variables in a class of distributions 12.00 -
12.30 a.m.
M. Dewess: The tail of distribution functions and its connection with the growth of its characteristic function
10 -
\
LIST OF PARTICIPANTS ARAT6, M., Res.
lnst. for Applied Computer Sci.,
Csalog~ny u.
30-32, PL 227,
BARTFAI, P., Math. u.
13-15,
Inst.
1536 Budapest, Hungary
of Hung. Acad. Sci., Re<anoda
1053 Budapest, Hungary
BERGSTRHM, H., Dept. Math. Chalmers Univ. of Techn. and Univ. of Goteborg, 40220 Goteborg, Sweden BRUINS, M.E., Joh. Verhulststraat 185, Amsterdam-ZI, The Netherlands CSAKI, E., Hath. u.
Inst. of Hung. Acad.
Sci., ReAltanoda
13-15,1053 Budapest, Hungary
CSIK6s, M., Nyisztor t~r 4/b, 2100 Godollo, Hungary CSIszAu, I., Math. u.
lnst. of Hung. Acad. Sci., ReAltanoda
13-15,1053 Budapest, Hungary
CSHRGO, S., Bolyai lnst. Jbzsef A. University, Aradi v~rtanuk
tere I, 6722 Szeged, Hungary
DAR6CZY, Z., Hath.
lnst. Kossuth L. University, PL
12,
4010 Debrecen, Hungary DEWESS, UONIKA, Dept. Hath., Karl Uarx Univ.,
701 Leipzig
GDR DUGUE, D., 24 Rue Jean Louis Sinet, 92330 Sceaux, France EBERL,
~.,T.,
Fleyer Str.
ERTSEY, 1., Hath.
122 c, 5800 Hagen, GFR
lnst. Kossuth L. University, PL
12,
4010 Debrecen, Hungary FEUER, tVA, Res. Inst. for Appl. Computer Sci., CsalogAny u.
30-32, PL 227,1536 Budapest, Hungary
FORTE, B., Dept. of Appl. Hath., Univ. of Waterloo, Waterloo, Ontario, Canada GERSTENKORN, T., Math.
Inst., Univ. of Lodz, ul.
Inzyniezska 8, 93569 Lodz, Polen GLEVITZKY, G., Math.
Inst. Kossuth L. University, Pf.
4010 Debrecen, Hungary
-
1I -
12,
GULyAs, 0., Inst. of Meteorology, Kitaibel P. u.l,
1024
Budapest, Hungary GYIRES, B., lIath.
Inst. Kossuth L. University, Pi.
12,
4010 Debrecen, Hungary JELITAI, A., Munklsotthon u.
34,
1043 Budapest, Hungary
KAC, M., Rockefeller Univ., New York, NY 10021, USA KATTI, S.K., Univ. of Missouri, 314 Math.
Sci. Building,
Columbia, HI 65201, USA KRAMLI, A., Res.
Inst.
Victor Hugo u. LAJK6, K., Hath.
for Compo and Automat.
24,
Sci.,
1132 Budapest, Hungary
Inst. Kossuth L. University, Pi.
12,
4010 Debrecen, Hungary LUKACS, E., 3727 Van Ness Str. NW, Washington, DC 2016, USA NEUETZ, T., Hath. u.
13-15,
Inst. of Hung. Acad.
Sci., Reliltanoda
1053 Budapest, Hungary
PANARETOS, J., 8 Gratesideias Str., Athens 504, Greece PAP, GY., Math.
Inst. Kossuth L. University, Pi.
12,
4010 Debrecen, Hungary PINCUS, R., Zentralinst. 39,
fur Hath. und Hech., Hohrenstr.
108 Berlin, GDR
PROHLE, T., Res.
Inst. for Appl. Computer Sci.,
Csalogliny u.
30-32, Pf.
RAISZ, P., VHrHsmarty u.
227,
1536 Budapest, Hungary
27, 3530 Hiskolc, Hungary
RAMACHANDRAN, B., Indian Stat.
Inst.,
7 SJSS Uarg, New
Delhi 110029, India REINITZ, JULIANNA, Hunklicsy u. RtVtSZ, P., Hath. u.
13-15,
I, 5350 l1iskolc, Hungary
Inst. of Hung. Acad.
Sci., Reliltanoda
1053 Budapest, Hungary
ROHATGI, V.K., Dept. Math., Bowling Green State Univ., Bowling Green, OH 43403, USA SARKADI, K., Hath. u.
13-15,
Inst. of Hung. Acad.
1053 Budapest, Hungary
-
12 -
Sci., Reliltanoda
SESHADRI, V., Dept. Math., McGill Univ., S05 Sherbrooke St. West, Montreal, H3A2K6 Canada SPIEGEL, G., National Planning Office, Angol u. 27, 1149 Budapest, Hungary STEUTEL, F.W., Dept. Math., Eindhoven Univ. of Technology P.O.Box 513, Eindhoven, The Netherlands SZtKELY, J.G., Hath. Inst. Eotvos L. University, Ufizeum krt. 6-S,
lOSS Budapest, Hungary
SZENTE, J., Res. lnst. for Appl. Computer Sci., CsalogAny u. 30-32, Pf. 227,1536 Budapest, Hungary
sztp,
A., Math. Inst. of Hung. Acad. Sci., ReAltanoda u.
13-15, 1053 Budapest, Hungary
TAR, L., Math. Inst. Kossuth L. Univ., Pf.
12,4010
Debrecen, Hungary TOMKO, J., Uath. lnst. Kossuth L. Univ., Pi.
12,4010
Debrecen, Hungary VlNCZE, I., Hath. lnst. of Hung. Acad. Sci., ReAltanoda u.
13-15, 1053 Budapest, Hungary
VIGASSY, J., Central Inst. for Phys. Res., Konkoly Thege fit, Pf. 49, 1525 Budapest, Hungary WOLFE, S.J., Dept. Hath. Univ. of Delaware, 501 Kirkbridge Office Building, Newark, Delaware 19711, USA XEKALAKI, EVDOKIA, Paxon Str.
IS, Athens 812, Greece
ZOLOTAREV, V.U., Steklov lnst. of Acad. Sci. of USSR, Vavi lova 42,
117333 Moscow, USSR
-
13 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
CHARACTERIZATIONS BY SUFFICIENT STATISTICS P.
BARTFAI
INTRODUCTION The result stating that the sample mean is a sufficient statistic for the location parameter of a family F(x-0)
of distribution functions only if
normal,
is well-known, but till now it has been proved
F(x)
is
only under strong restrictions. For e.g. KAGAN-LINNIK-RAO ([IJ, Theorems 8.5.3 and 8.5.4) assume the existence of the density of the sample mean. Now we shall show that this statement as well as the analogous statement for the scale parameter is true without any condition. I.
THE MEAN IS SUFFICIENT STATISTIC FOR THE
LOCATION PARAMETER THEOREM I. Let distribution fU1lction
X 1 'X 2 " " ' X n F(x-O).
e
sufficient statistic for
-
Then
iff
15 -
be i.i.d.
r.v.-s with
I ;:;(X I +X 2 + •• • +X n )
F(x)
is normal.
is
PROOF. Consider the n-dimensional sample space (Rn,B(n),P a ) where B(n) is the set of the n-dimensional Borel sets, P a distribution function Ea
is
the measure generated by the
F(X I -a)F(x 2 -a) ... F(Xn-a). Write
for the expectation with respect to x1+···+x n
Pa. Let
r(~)
and
n
c then the equality
Po-a.e. and h does not depend on it(T-a) e and taking an expectaa. MUltiplying it by
holds for every
a
tion we obtain ( 1)
The expectation on the left hand side does Rot depend on a
because the set
shift by a vector
C
is invariant with respect to a
(a,a, ... ,a)
and so
f (t) •
Therefore the right hand side of (I)
is equal to
f(t),
too, or, which is the same (2)
f(t). Introduce the distribution function
our aim (4) we can assume that
-
16 -
(considering
EO(lc) > 0)
(3)
it is really a distribution function because
f(O) f(O)
I.
The characteristic function of
f(t)
(2) ,
is, according to
therefore, using the Unicity Theorem of G 8 (z)
the characteristic functions,
i.e.
8,
=
G 8 (z)
cannot depend on
G(z).
Let us choose the value of W 1 +W 2 +· •• +wn
8
in (3)
8
=
it by
--
n
=
interchanged because
0 $ h $
1
a.e., and we get for
the inner integral that 00
••• J
f
E (E (I T
T
C
E (l c !8(W)+T)dF(W I ) ••• dF(W) T n !8(w)))
which implies
A
z
(~:
T(~)
< z},
we obtain
f ••. f (4)
=
(w l ,w 2 ' ••• ,w )ER) and integrate n dF(WI) ••• dF(W n ). The two integrals can be (w
n
Let
8 (w)
A
h(T)dF1 ••• dF n
z
-
17 -
Consider the r.v.-s
that to
X I 'X 2 ' ••• 'X n again (4) means XI+ ••• +Xn are independent with respect PO. This relation leads us to the well-known func-
X2 -X I
and
~(t)
tional equation for the characteristic function of
XI
~(t+s)~(t-s)
£}
o
r .... oo
for any (iii)
sup n
~
n
£
(S)
> 0, < +00.
REMARK. The conditions for the weak convergence of
n
(r)
-
x
and
PROOF. Let
p ( x, L
n
v(r)
n (r) x )
'1'0.
are only measurable.
be the realvalued bounded linear
functional corresponding to class
(i)-(iii) are sufficient 1'kn also if the mappings ~n
~n
and considered on the
By
( 1 .3)
a bounded linear functional is defined. By the transformation (I. 4)
x=n
(r)
x
we get
L (r) (f)
n
Now assume that (i)-(iii) hold. Then by (i) converges to a bounded linear functional '1'0
as
n -
+00.
Using the form (1.4) for
further get
-
23 -
L(r)(f)
L(r)~f)
on
L~r)(f)
we
IL n (f)-L(r)(f)! n
I
(). 5)
S
sup
If(x)-f(y) !~n(S)+2 suplf(x)
p(X,y)SE
xEs
J
1
p
«r) r
x,v
1t
» x
The second term on the right hand side tends to n -
r
+00,
-
+00,
~n(dx). E
as
0
according to (ii). The first term is
arbitrarily small for sufficiently small
since
E,
is
f
uniformly continuous. Hence
o.
lim lim suplL (f)-L(r)(f)1 n n r ....co n .... oo
( I .6)
We now consider
IL
(r ) (r ) 1 (f)-L 2
(\.7) +IL
(r ) l(f)_L
Inn
Using the fact that n -
+00
L(r)
(f)
(£)1
S
IL
(r
(f)I+IL (f)- L
n
L (r) (f)
) 1
(r (f)-L
n
(f)
1
+
(r ) (r ) (r ) 2 (f)I+IL 2 (f)-L 2 (f) 1
n
n
L (r) (f)
converges to
n
) 1
as
and also using (1.6), we find by (1.7) that is Cauchy covergent and hence convergent lim
L(r)(f)
=
L(f)
r ... +oo
where
L
is a bounded linear functional on ~
hence determines a measure find that +00 {~} 1 n n=
{L(n) (f)}
on ~.
and
S. By (1.6) we than
converges to
converges weakly to
~O
L(f)
and thus that
This measure is a-smooth
according to Alexandroff's second theorem since the are
~n
a-smooth. Before proving the necessity of the conditions (i)-
-(iii), we verify the statement in the remark. Clearly, so far we have only required measurability of the -
24 -
mappings
1l
(r)
and
p(x,v(r)ll(r)x). Hence the state-
x -
ment in the remark is true. We now prove the necessity of the conditions
(i)-
-(iii) under the conditions on the mappings given there. Since
1l
(r)
+00 {].In} n= 1
Hence
is continuous the weak convergence of
. l '1es t h e wea k 1mp
(i)
is necessary.
convergence
is equal to
on
0
to the
Jg £ [p(x,v r
1l
(r)
Jg £ [p(x,v r Now
1l
0
[£+00],
x)11l
(r)
n
s
and the converimplies
(dx)
x)11l(dx).
(r -
as
£e: r
n
(dx)
0,
r .... + oo
lim sup k 6 x II (dx) n f n r n n .... + oo 116 xiISe: r
lim
lim sup k
r .... + oo
n ....+ oo
n f
116 xl r
2
II
n
(dx)
0,
O.
For the proof of this theorem we need several lemmas.
-
30 -
LEMMA 2.1. Let
~2
OIl
such that
A=~*V. If
A({x:llt. xII
>
r
then there exists ~{x:llt.
r
We may choose etrical,
x
r
>
xU
and assume that 1-e: }. Then r
°
1-e:
put
r
.
r
r
f~(K-x)v(dx)
A(K)
>
0,
xr· If we get
if and only if
{x:Ut. xII ~ e:} r E = {x:~(K-x»
V(E c
1, i.e. vee)
0, which
is not empty. Thus we may choose
> 1-e: r , i.e. ) = t. x+t. x • r r r
=
K =
Further put
and then by the definition of
r
-K=K
)
1-E
and
r
>
S E}
].l{y:n6 yn
r
r
yn S E}. ].l(K+x
1-2E
r
r
R
we hence have
r
.
We shall use the function x into
> 1-E
)
expC-nxn 2 ) from
12
and now give some properties of this function.
LEMMA 2.2.
We have
!exp(-nxn 2 )-expC-nx+tn 2 )-2x.t exp(-nxI1 2 )! S a(n)ntn 2
(i)
with· finite a(n) g(x,t)
=
ntn S n < +00. Further
for
2 2 2 2exp(-nxn )-exp(-nx-tn )-exp(-nx+tn )
satisfies the inequalities (ii)
!g(x,t)!
(iii)
g(x,t)
1
ntn S I
for
for
for
c 1 (n)
!g(x,t)1 S c 2 lit
(i v)
(v)
for all
x
and 1
t,
I xn S 7; nn tn
with
n>o,ntn 2n,
for
of
~
S 2
x
and
2
with a positive number
(independent
t), ~
g(x,t)
nxn S
I
±,
c3ntll
ntn S
(independent of
x
2
± and
with a positive number t).
- 32 -
c3
PROOF.
II xII
Observing that 2
-lIx+tll
IIItll2
for
II til
-to
=
::; n
we
get
2
-II til
-2x·t,
::; n 2 + 211xlln
+ 2x·tl
(i)
by the help of
exp-[ 1It11 2 +2x.t].
expansion of obvious and
2
(iv)
The
is obtained by
For the proof of
(iii)
the Taylor
inequality
(i)
regarded
(ii) for
is t
and
we write
2 2 exp (-lIxli )[ 2-exp(-1It1i +2Ix.t!)-
g(x,t) (2.5)
-exp (-II tIl 2 -2! x· t I)]
and observe that for
II xii
~ exp (
1
4"
::; 1
n,
II til II til
2
~
2,
-2!x.t!
i
Then by
n. 1
~
2
-T6'l )(2-2exp(- "2 n )) =
IItIl2+!x.t~ ~
n2
(2.5) we get
c 1 (n).
g(x,t)
For the proof of
we write
g(x,t)
By Taylor's
=
2exp{-CIixli
formula we
cosh(2!Xotl)-
and
get,
2
+11 til
for
2
)}[exp(lltli
II xii
::; -21 IItll2 cosh
thus
and
-
33 -
n2
::;
±, 1
"2 '
2
)-
II til
::; 2 '
~
(v)
LEMMA 2.3.
{A
Let
} +00
(r= 1 ,2, •.. )
n,r n=1 of probability measures on i 2 tending to
positive integers
lim sup k
lim
(i)
r--+ oo
n"'+ oo
lim sup k
lim
(ii)
r"'+ oo
n"'+ oo
lim sup k
lim
(iii)
r"'+ oo for
n"'+ oo
a
n -
+00.
as
+00
J
A
J
t.
n lit. yl>e: n,r r n lit. yiISe: r r
YA
sequence of If
0,
(dy)
(dy)O
0,
n,r
U,\ yl12A
J n lit. ylSe: r
(dy)
r
n,r
n(dy)
°
°
A
lit. yll>n n,r
n ..... + oo
r"'+ oo
"'k
J
lim sup
lim
r
n > 0.
any
PROOF. Let put
{k} n
e: > 0, then
any
(i v)
for
and
be sequences
*0 A n,r
e. We have the identity
e-A
(2.6)
be the unit probability measure and
e
"'k
k
n
n,r
n !:
-I
j=O
A'~j (e-A n,r
n,r
).
By this identity and the properties of convolutions we get, applying (i) of Lemma 2.2 i'k
J[ l-exp(-IIt. r Y112)] An, n(dy) S r -I
k n
!:
j=O
suplf{exp(-IIt. xii
2
r
I
-
34 -
2 )-exp(-IIt. (x+y) II )}A r
n,r
(dy)l:::;
S k
f
nll6. yll>n
A
r
n,r
+ 2k II f 6. y A Cdy) II n 116. yllSn r n-r
(dy)
+
r
+ k a(n)fn6.
n
r
yR 2 An,r (dy).
By (i)-(iii) the right hand side of the last inequality
0
tends to (iv)
n -
as
+~
and
r
+~
-
in this order. Then
follows by the right hand side of the first inequal-
ity. Applying Lemma 2.3 and Theorem 1.1 {~n}
to the sequence
of probability measures we find that the conditions
(i)-(iv)
in Theorem 2.2 are sufficient for the weak ":k n convergence of ~n We have already remarked that (i) is necessary for this covergence. that also
It remains to prove
(ii)-(iv) are necessary conditions. We shall
first prove this for symmetrical probability measures ~
. Using the identity (2.6) we get according to the n symmetry of ~n
f exp(-II6. (x-t)1I2)[e(dt)-~ r
':k
n
n(dt)]
f{[exp(-II6. xII 2 )-exp(-II6. (x-t)1I 2 )lll "'k n(dt) r r n 2
f{exp(-IIL\ xii r
1 -2
f
g(6. x,6. r r
with the function
g
2
)-exp(-II6. (x+t) II r
,~k
n
n(dt)
t)~ n (dt) considered in Lemma 2.3.
Hence forming the convolution of the signed measure
)}~
*k n
e-~n
exp(-I6. x1l2) r
and applying (2.6) we get
the identity (2.7)
f
g(6. x,6. t)~ r
r
*k kn-I n(dt) = ~ n j-O
f
f
35 -
h. (6. x,6. t) 11 J
r
r
n
(dt)
with
with
h. (!:J. x,!:J.
(2.8)
]
We use
r
r
t)
= fg[!:J.
the identity
zeroelement.
(2.7)
r
!:J. t] / ' j (dy). r n
(x-y),
in the case when
x
By our assumption the sequence
converges weakly.
Then by Theorem
lim r-++ oo
0
to
r
as
Hp.nce
lim sup lln n ({x: II !:J.rxll n ..... + oo
r > O.
for any
for a
have the
-
We observe that
A
if
+00
A
sequence
lim sup
A({x:lI!:J. xII r
/'j({x:lI[\ xII
n
r=r(£)
clearly r=r(£)).
n
r
£ > 0
E)
= 0
for any
there exists
r
> n
and all
(2.10)
holds
O. Thus we find that the conditions
for any (iv)
0
,"k
(repeated limit), since (ii) and
n(dt)~
n 2
The left hand side tends to 0
correspondingly
E. Using these estimations we
2J[ I-exp(-II~ til
to
r
(n S I)'
S n
for sufficiently small obtain by
(and I
in Theorem 2.2 are satisfied.
is obvious since
Clearly (iii)
is symmetrical.
We shall now remove the restriction that the are symmetrical. Then let in the sense
)j (E)=ll (-E)
in
II ;')j
J/,2. Then
n
for any measurable set
n
n
is symmetrical. Let
n
,',k
*
~
,',k
n = (ll
weakly to
ll.
weakly to
ll*)j. Hence the sequence
the conditions (2. I I)
lim r-'+ oo
(2. 12)
lim r-++ oo
Then
II
n
n
(ii) and lim sup k n-'+ oo
n
(iv)
n
*
,',k
{ll
n ,',k
ll) n
{ll *)j}
n n in Theorem 2.2,
J
II ",)j (dy) n n
J
II~
nn~ yll>E
n}
n
exists
converges
0
r
lim sup k n n-++ oo
r=r(E)
E
converge
satisfies
Ut:.ryIISE
r
xii
2
II ;'ll (dy) n n
The relation (2. II) holds if pnd only if to any there
lln
be the "conjugate" of
such that
\
-
37 -
O.
E > 0
(2. 13)
f ~ *~ 06 yO>£ n n
£ n,r
r n,r
=x
n
r(£». Note that
n,r
x
n,r
Hence
lim sup k f v (dx) n_+oo n06 xO>£ n,r
o
r
> o.
£
for all
£
kn
Put
f 6 xv (dx), 06 xO:s:£ r n,r r
A
n,r
For any
v
n,r
(. +m
£' (0 < £' < £) Om
n,r
O:s:
n,r
)•
we get
f
06 xn:s:£'
6 xv r
r
n,r
(dx)O+£
f
r
Hence (2.14) implies (2.15)
lim
lim sup Om
n,r
n
O.
Then it follows by (2.14) (2. 16)
lim
lim sup k
n
f
n6 xD>£ , r
Further -
38 -
v
(dx)
n,r
v
06 xO>£' n,r
O.
(dx).
II J t. r lit. xII::;£: r
X" n,r (dx)1I
t. xv (dx+m ) II II J r n,r n,r lit. xII::;£: r
II J (t. y-m )v (dy)lI::; lit. y-m II::;£: r n,r n,r r n,r
::;
J
lit. yll::;£: r
t. yv (dy)-m 11+ r n,r n,r
+2(£:+lIm
J
II)
n,r Note that the first
lim
lim sup k
r-+ oo
n-+ oo
Observing that
o
"
n,r
Hence
*"
n,r
=p *p n n
r-+oo
Regarding
J
J J nllt. (x)+t. yll::;£: r r
~ lim sup lim sup
(2.16)
n-+oo
and
(2.15)
we obtain by
n lit. xll E)) n,x x r ..... + oo
(2.19)
>
o
Land L be the realvalued n n, x bounded linear functionals corresponding to lln and for any
A
E
O. Let
respectively and considered on
p~~jection
n(x)
of
£2
onto II
n
R(x)
'1'0. For the we have
( n (x)-I • )
since As in the proof of Theorem 1.1 the measure II (n(x)-I.) determines a realvalued bounded linear funcn tional L(x)(f) on '1'0. According to Theorem I. I and the proof of this theorem we have lim lim sup
x-+~
n-+~
(f) -L (r) (f) 1
1L
n
O.
n
Applying the inequality (1.5) to the difference
-L(x)(f) n
and regarding (2.14) we obtain lim lim sup
x-+~
n-+~
!L
o.
(f) -L (x) (f) 1
n,x
n
The two last inequalities give lim lim sup r~+oo
n ..... +co
IL n (f)-L n, x (f)1
(2.20) lim lim sup r .... +oo
n ..... + oo
IL(f)-L n,x (f)1
- 40 -
o
L
n,x
(£)-
is the functional belonging to the
L (f)
where measure
being the weak limit of
II ,
"'k
= lln n{.+k
n
"'k n II
(m
n,r
-x
n,r
)}.
converges weakly to
n
But according to as
n -
+00,
-x
U
r -
a-smooth ,',k
A n = Now n,r A"·'·k n (2.20) n,r
+00.
This is only
possible if
(2.21)
lim lim sup k
n
Um
If this condition holds,
n,r
n,r
o.
then it follows
this relation holds true if we change i.e.
for
lln
we find that
from
>
E
An,r =ll n
that
that
into lln' n,r Proceeding then as above
instead of 'J n,r (2.21) holds with
Um
n,r
A
i. e.
U =0
lim lim sup k U J I:::, Xll (dx) U n . . . . + oo n UI:::, XU$E r n r for any
(2.16)
o
O. At last we find by (2.18) applied to
lim lim sup k J UI:::, xU 2ll (dx) r-+ oo n-+ oo nUl:::, XU$E r n
0
r
>
O. Thus we have proved that the conditions
for any
E
(i)-(iv)
in Theorem 2.2 are necessary,
Consider now an infinitely divisible probability "'n for all positive inmeasure II on £2. Then ll=lln tegers to a
n.
It follows by Theorem 2.2 that
a-smooth
J
a-finite measure
UxU
2
q(dx)
< +00,
UxU $1
J
n xn >E
q(dx)
~
O. Further
projected measures.
is determined by its
It can easily be proved that the
Gaussian invariants of the projected measures for finite pr0i!ctions have the Gaussian representations given in Theorem 2.1 if we change
=
weak limit of
n~n'
{~n}
q
being the sequence of
~2, and consider
probability measures on elements in
into the corresponding
q
x
and
y
as
~2. The fact that the weak limit of a
convolution product
~
*k n
n in
~
2
is infinitely divisible
can be deduced from the well-known theorems about weak convergence of such products in
R. We have then to apply
Cramer-Wold's theorem. REFERENCES [I]
A.D. Alexandroff, Additive set-functions in abstract spaces, a) Mat. b) Mat.
Sbornik, 9(1941), 563-628, c) Mat.
13(1943), [2]
H.
Sbornik,
169-238.
Bergstrom, Limit Theorems for Convolutions,
Almqvist & Wiksell, New YorklLondon,
H.
Sbornik, 8(1940),307-348,
Stockholm, John Wiley & Sons,
1963.
Bergstrom
Dept. Math., Chalmers Univ. of Techn. and University of Goteborg 40220 Goteborg, Sweden
-
42 -
COLLOQUIA HATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ON SOME DISTRIBUTIONS CONCERNING MAXIMUM AND MINIMUM OF A WIENER PROCESS E.
csAKI
1.
INTRODUCTION
Let
w(t)
be a standard Wiener process, w(O)=O
and put ( 1 • 1)
M+(t)
w(u),
max O~u~t
( 1 .2)
M- (t) = - min
w(u),
O~u~t
(1. 3)
M(t)
max
I w(u) I
max (M+ (t), M- (t».
O~u~t
It is known
(see e.8. RtNYI [8J), that the use of
theta functions and certain identities between them lends itself particularly well to investigate the distribution of
M(t). For theta functions and identities a standard
reference is e.g. MAGNUS-OBERHETTINGER [7J. The definition of theta functions used in this paper is given below. \- 43 -
00
(1.4)
~
1+2
-&O(V,T)
2
(-I)n e iltTn cos 2nltv,
n=1 00
( 1 .5)
( - I)n e iltT(n+I/2)2 S1n . (2 n+ I) ltV,
~
2
-&I(V,T)
n=O 00
( 1 .6)
2
-&2(V,T)
e iltT(n+I/2)2 co s (2 n+ I)
~
It
v
n=O 00
(1.7)
1+2
-&3(V,T)
~
e
iltTn
2 cos 2nltv.
n=1 In this paper we determine the joint distribution of
w(t):
the following characteristics of
(I .8)
R(t)
(1.9)
Q(t) R(t)
+ M (t) R"(tT
(0
for any £ > 0 x,yER m we have
semicontinuous if
0
such that for
there exist a
s(y)C{vERn:dist (v,S(x» < £} whenever dist (x,y) < 6. n m Also, it is said to be concave if for each 0 $ a $ m and x,yER, as(x)+()-a)S(y)~S(ax+()-a)y). and denote the origin of R n Let 0 = (0, ••• ,0) Bn
we write simply centered in
for the closed unit ball in cERn and a positive number
o. For
=
B (c,p) n
~,f'(~)
tion
f
c
and radius
p. For a
denotes the first derivative of a func-
and for
t, uERn, (t,
product. The following result number of the T
p
C+pB n
is then the ball with center real
Rn
u)
(with
stands for their inner n+)
replaced by the
(n-I)-dimensional faces) holds true if
is any (not necessarily bounded) convex polyhedron,
but for our purpose a simplex suffices. LEMMA 10. If
T
is any simplex in
Rn, and
-vol [TnB (c,p)], then one can find simplices n n n-) and constants Bn ,a) , ... ,a n+ ) ••• ,Tn +) in R with the functions
A
n
(p)
n-)
A
n-
)
,
.(p)=vol
~
n-
)[ T.npB ~
A (p)=
n
T), •••
1
so that
we have
=
(3. I)
Here the value of
A
n-
)
,
.(p)
~
equals to the
(n-)-
dimensional volume of the intersection between the i-th -
58 -
face of P
and the
T
(n-l)-dimensiona1 ball of radius
centered at the projection
the original
of the center
c.
~
of
c
n-dimensional ball on the supporting
(n-l)-dimensiona1 hyperplane of the
i-th face.
Furth-
ermore, dist
[
ex.
~
n
if
(c,c.), ~
c.-c ~
is a non-negative multiple of
-dist
n
if
(c,c.), ~
c.-c ~
u.
is the normal vector of the
~
pointing outward from
~
is a non-negative mUltiple of
where
u.
ui '
i-th face of
T,
At last,
T.
if
Bn
=[
D,
n
where
T
if
n
cET,
is the n-dimensional spatial angle of the n cone formed by the rays issued from c, having an T
intersection with if
T
of positive length. In particular, T then B =vo1 B n = n n
is an inner point of
c
(Ill) n /
r (~
+ 1) •
PROOF. Without any loss of generality we suppose that the
center of the ball is the origin, i.e. n
c=o,
Bn(C,P)=PB .
Let
uERn
be a unit vector and
KeRn
be a compact ,
convex set. Assume that the two supporting hyperplanes of K
which are ortogona1 to
with
K
of less than
seen that the function is differentiable and,
n-)
u
have intersection figures dimension. Then it is easily
f(~)=vo1n[Kn{tERn:(t,U)
for all
J - 59 -
~,f'(~)
=
:;;
0]
= vol that
nif
(j f.k)
I [ Kn ( t : ( t ,
U )
= E;}].
i t
f
0
11 0 W s d ire c t 1 y
u I ' . . . ,umER are unit vectors with such that the intersection figures of
those of its
supporting hyperplanes
to some of the dimensions,
U. ' ~
s
then for
A (E; . ) ~
= (t : (
t,
lying orthogonally
are of less then
(i=I, ••• ,m)
F:R m -
the function
= vol n [ KnA ( E; I ) n.
F ( E; I ' . . . , E; m )
(3.2)
where
Fro m her e
n
~
~
vol n _ I [ KnA ( E; I ) n.
defined by
. . nA ( E; n )] ,
(i=1 , . . . ,m), we have
E;.} eRn
:0;
U .)
R
n-I
. . nA ( E; i-I ) n
(3.3)
n (t
: ( t,
Now we in
(3.3)
U .) ~
.
cla~m
= E; .} ~
nA (E;.~+ I) n ... nA (E; n ) ]
that
t
h
e
.
part~al
..
der~vat~ves
are continuous on the whole
the notations
D.(E;I, . . . ,E; ]
=(t:( t , u / =E;j}
and
(i=I, . . . ,m).
)=A(E;.), m ] K(E;I"" ,E;m)=K,
Indeed,
Rm.
E·(E;I, .. ·,E;
] the
of
~
m fun ct ion
~ith )= 0F
~ ~
can be considered by
(3.3)
Lebesgue measure of the
as
the
intersection of
concave and upper semicontinuous ••• ,D.
~-
1('),
D.
(n-I)-dimensional
functions
in [2J)
by
set-valued D I ( · ) , · ..
I, . . . ,D
m
(·),E.(·) ~
and
K(·).
function on
the well-known Brunn-Minkowski theorem
one can write
of
~
u
:0; ]
n+1
-
~
(i=I, . . . ,n+I).
Ci.=(U.,o':')
>
~
i=1
a. ---~-Ivol p
n+
n-
I[ Bnn{t:(
a. .....2} P
n-
t,u'> ~
=
~}n p
1
I[ PBnn{t:(
-
and by
0':
Let
~
ui ' .',
(the origin of
o
~
to
61
-
t,u'> ~
Ci .} ~
n
By
n+1
n n {t:(t.u'>:S ct,}] j=1
]
T':npB n
Observe now that ~
Ip2-ct~
R
• w1th
H . (0 1:)
~
O:S P
-
< Ict.l·
if
n-I
.~
~
~
=0
ER
n-i
(i=I ••••• n+l)
wise arbitrary). for the choice
= vol
n-
V
T.
~
2 + B n-I ] I[T.n (p 2 -ct.)
Hence. the case
~
~
of formula
c=o
= H. ( ~
T": ) ~
for H. :E. ~
~
-
(otherwe have
(i=I.2 ••••• n+I).
(3. I) follows with
lim vol [TnpB n ]. p"O n i. e ••
if where cone
T
n
is the
oET.
n-dimensional spatial angle of the
(O.oo)XT. But then the lemma also follows in the
stated generality. Viewing now the volume as a function of the radius we immediately have the following COROLLARY.
A (p) n
is
0': ~
(n-I)-dimensional affin sub-
in the
~
is a ball of center
~
= dist n (0.0":). and is p ~ Ict.1 ~ ~ Hence. considering an isometry
E~
space
E~npBn
where
~
and radius
here can be written in the
~
T~n(E~npBn)
form
]
d(n)
uously differentiable.
- 62 -
times contin-
PROOF.
For
continuity of some
k
~
2
t h at t h e f
n=l, AI (p),
is true.
, (d (k-I ) Hk I '
unct~ons
atives of order (~d(k»
and this
the claim is only the
the assertion holds for .
It follows
~.e.
d(n)=O,
-
,~
of
d(k-I)
then from
(
(3.1)
Suppose that for
i= I, . . . ,k+ I
)
,
.
der~v-
the
exist and continuous.
Ak _ 1 i ' ,
that
This means
k-I.
is
Ak
d(k-I)+I
times continuously differentiable over the set
(O,oo)'-.{al, .. ·,a k + I }· and to prove that
~
For ~s
Ak
~ ai'
Ak_l,i(
l '(~ 2 -a i2 ) + )
:= 0,
times continuously
d(k)
differentiable also in the points
it is
enough to show that
(3.4)
lim ~"a
But, by
(3. I) again,
constant
(i=I, . . . ,k+I).
°
. ~
ck_l,i'
Ak_l,i
if
h )
k-I ck_l,i
=
T
is small enough.
T
with some
Hence,
to show
(3.4) is the same thing as checking lim
(3.5)
(V=d(k)-I=[I] - I ) .
°
~" a
But
[~]-I 2
~
p.
j=1
where the follows.
]
(U
(~
k-l
2 -2- - j -a )
P.'-s are some polynomials, whence ]
(3.5)
By induction the Corollary is proved.
Taking the simplex
S
of Section 2 as an example
n
we see that the Corollary cannot, ed.
2
in general, be improv-
On the other hand, we saw that we have troubles with p's only, where the ball
differentiability in those
knocks against different dimensional
(n-I ,n-2, ••• ,n)
faces of the boundary of the
In fact,
easy to prove that
A
n
(p)
I-
simplex.
it is
is piecewise analytic on 63 -
(0,00) •
4. COMMENTS ON Because of
V (x) n
(2.2)
and Lemma 10,
V (x) n
has the
following form
V
n
(x)
n!A (p(x» n
(4. I)
n+1
n!pn(x>{S - ~ n
i=1
,
An- I , 3=··· =A n- I ,n+ I
where I
=--; A =A lIn n-I, I n - I =nn/2/ r
(I
2
+1), since
is easily seen that follows then, points of the
c
S
is an inner point of
n
sn
and
n S
It
n
do not have obtuse angles.
that the projections of
.
cn
(n-I)-dimensional faces,
will be inner
the projections
of these projections will be inner points of the -dimensional faces of the
It
(n-2)-
(n-I)-dimensional faces,
etc.
A .(/(~2-a~)+) by (4.1), the n-I ,~ ~ corresponding constants S I . will again be the So, when evaluating
n-
volume of the
,~
(n-I)-dimensional unit ball
and all the corresponding constants
i=1 , ... ,n+l,
a ..
(j=I, ... ,n) ~J will be positive, and this phenomenon is persistent with the decrease of dimension.
In this sense the recursion
in (4. I)
But one also notes that in
is "homogeneous".
the second step it will not be true that two of the
a's is the same and the rest is again the same
(i.e.
the ball reaches two faces at the same time, and, a bit later, it reaches the other faces again at the same time).
This "regularity"
disappears after the step, as
seen starting out from three dimensions.
-
64 -
Much work has been done to compile tables of percen2 wand similar statistics, in partic-
tage points for
n
ular by STEPHENS. in KNOTT [3J.
A survey and comparison can be found
In fact,
the most accurate.
Knott's results are proved to be
All these results,
on some kind of approximation of formula
Lemma
statistics, Knott.
Vn(x).
are based
In principle,
gives the possibility of the exact tab-
(4.1)
ulation.
tables,
10 is also applicable
e.g.,
M2n
for the
for other similar
statistic of Durbin and
seems to be accessible on a computer.
n=20
Unfortunately,
our computer facilities here are not
adequate at present to do this work. REFERENCES [I
J
S.
Csorg5, On an asymptotic expansion for the von Mises w2 statistic, Acta. Sci. Math. (Szeged), 38(1976),45-67.
[2J
H.
Hadwiger,
Vorlesungen
und Isoperimetrie,
Heidelberg, [3J
M.
Knott,
uber Inhalt,
Springer,
Berlin-Gottingen-
1957.
The distribution of the Cramer-von Mises
statistic for small sample sizes, J. Soc.,
S.
"-
Ser.
Oberflache
B.,
36(1974),
Royal Stat.
430-438.
Csorgo
and L.
Stacho
Bolyai Inst.
of Jozsef A.
Aradi vertanuk tere
I,
University
6722 Szeged, Hungary
-
65 -
,
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY).
1977.
NONNEGATIVE INFORMATION FUNCTIONS Z. DAR6CZY - GY. HAKSA
I.
INTRODUCTION
The notion of information functions has been introduced by Z. DAR6CZY [3J. A summary of the investigations concerning information functions can be found in the book [IJ by J.
ACZ~L and Z. DAR6CZY.
DEFINITION. A real-valued function the closed interval
and
f(O)
f
( I .2)
= f(I),
I f(-Z)
=
I
satisfies the functional equation x
f (x) + (I-x) f (-y-)
f(y) + (I-y) f(-I- ) -y
I-x
for all (1. 3)
is said to be an information
I]
if
function
(1. I)
[0,
defined on
f
(x,y)ED, where D
{(x,y): 0 S x
, •••
,L'»',
a
x*
where the components of and independent of
z
are mutually independent
which has a continuous distribu-
tions having zero mean and a finite third absolute moment.
is a non negative mixing constant and
L'>
may take values +1
or
We shall compute
-I.
Pitman efficiency of the test based on that based on
T j • Let
a in ,
bi~
aiO'
~
in particular,
are bounded constants which,
(i=I, ••• ,p)
a.
the
E .. ~J
relative to
T.
~
respectively be
HL'>' effective mean under
the effective mean under
and effective standard deviation under
HO
of
HO
T .• ~
Towards this, note that
E (x .x .1 H ~
x.
where
]
1/2 2 2 a.a.1:I (J In
)
x.]
and
~
n
is the variance of
~
]
are two components of Z.
aJ/.n
In
and
(J 2
It follows that p p
L'>2(J2
(4.2)
x
~ ~
p (p-I )
Uj
For the non null means of
2 2
a .a .' ~
]
and
T2
T3
we have the
following lemma.
LEMMA 4. I. If
f.
~
sities of the components of ponding distribution
x'"
functions)
(and
f'.'
ii)
tegrable with respect to 2 as f i (x) .... 0 Ix! .... d,
then under
exists,
F.
~
the corres-
satisfy
i)
~
the marginal den-
(i=I, . . . ,p)
is continuous and uniformly in-
HL'>'
-
108 -
Fi ,
(i=I, . . . ,p),
(I
(2)
RSa
T
In
:::;; p)
p (p-I )
2 2 2
S=~~
'f: k
2
2 a S'
--+ 2P- 1
where
:::;; j
2
-1/2) + 0 (n
2
a.a.6. Jf.dxJf.dx ~ ] ~ ]
i'f:j
PROOF. We shall prove special case and
(3)
since (I) follows as a
(2) becomes a simple corollary of
(3).
Consider and
= 2P(X*i ,Q.
\
=2 J z
m
where Z.
G
J z,Q.
are concordant)
X
-In
I/4,
=
HO:l:l
O. AIso, let
0
against the alternative
(x I k ' X 2 k ' •.. , X pk) ,
a random sample of size population. Let
R. = ~
n ~
denote the vector
~n
n
on the
i-th compon-
(i=I, ... ,p). Then, we have the following theorem.
Ez2 -I
is normal and is greater
is not normal. Thus
ymptotically optimal when the
=
F.
~
TI
is as-
are normal.
REFERENCES [I]\T.W. Anderson.
Introduction to multi-variate
analysis. John Wiley. New York.
[2]
C.B. Bell - P.J.
1958.
Smith. Some nonparameteric tests
for the multi-variate goodness of fit. multi-sample independence and symmetry problems. Multi-variate Analysis II. Ed.
New York. [3]
J.R.
P.R. Krishnaiah. Academic Press.
1969.
Blum - J. Kiefer - M. Rosenblatt. Distribution-
free tests of independence based on sample distribution function. Ann. Math. Statist .• 32(1961), 485-498. [4]
S. Bhuchongkul. A class of nonparametric tests for independence in bivariate populations. Ann. Math. Statist .• 35(1964).
138-149.
-
119 -
[5]
N. Blomquist, On a measure of dependence between two random variables, Ann. Math.
Statist.,
21(1950),
593-600. [6]
P.C. Consul, On the exact distribution of the likelihood ratio criteria for testing independence of sets of variates under null hypothesis, Ann. Math.
[7]
R.C. Elandt, A Acad.
[8]
38(1967),1160-1169.
Statist.,
Polon.
nonpa~ametric
Sci., Ser.
Sci.
test of tendency, Bull. Biol., 5(1957),
187-190.
R.C. Elandt, Exact and approximate power of the nonparametric test of tendency, Ann. Math.
Statist.,
33(1962), 471-481. [9]
D.A.S. Fraser, Nonparametric methods in statistics, Wiley, New York,
1957.
[10] D.V. Gokhale, On asymptotic efficiencies of a class of rank tests for independence of two variables, Ann.
[11]
w.
Inst.
Statist.
Math.,
20(1968), 255-261.
Hoeffding, A nonparametric test of independence,
Ann.
Math.
19(1948),546-557.
Statist.,
[12] W. Hoeffding, A class of statistics with asymptotically normal distributions, Ann. Math. Statist., 19(1948), 293-325. [13] J. Hajek, Locally most powerful rank test of independence, Studies in Math. Statist., Theory and Applications, Akad. Kiad6, Budapest, [14] J. Hajek - Z.
1968, 45-51.
Sidak, Theory of rank tests, Academic
Press, New York,
1967.
[15] D.N. Lawley, The estimation of factor loadings by the method of maximum likelihood, Proc. Roy. Soc. Edinburgh, 40(1940), 64-82.
-
120 -
[16J E.L. Lehmann, Some concepts of dependence, Ann. Math.
Statist.,
37(1966),
1137-1153.
[17J D.F. Morrison, Multi-variate statistical methods, McGraw Hill, New York,
1967.
[18J G.E. Noether, Elements of nonparametric statistics, Wiley, New York, [19J M.L. Puri - P.K.
1967. Sen - D.V. Gokhale, On a class of
rank order tests for independence in multi-variate distributions, Sankhya Ser. A., 32(1970), 271-198. Addendum. HO
The asymptotic normality of
under
follows from a result of Cramer (Mathematical Methods
in Statistics, Princetion Univ. Press,
1946, pp.
254-255).
Its asymptotic normality under all the hypothesis follows from a str~ht forward extension of a result of Cramer (1946, p. 366) pertaining to the asymptotic normality of functions of sample moments. Acknowledgements.
The authors thank the referee, and
Professors Bartfai and Vincze for a critical reading of the manuscript.
z.
Govindarajulu
Dept.
Stat., Univ. of Kentucky
857 Patterson Office Tower Lexington, KY 40506, USA A.P. Gore Maharasta Association for Cultivation of Science Poona, India
-
121 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ON A GENERALIZATION OF STIRLING'S NUMBERS OF THE FIRST KIND B.
GYIRES
INTRODUCTION
(
In this paper quantities are defined which can be considered as a generalization of Stirling's numbers of the first kind
([IJ,
Chapter IV).
It will be shown that
these quantities and their generating functions certain partial difference equations Sec.
(Sec.
satisfy
1 and 2).
In
3 we give an explicite representation of these qua-
ntities and of their generating functions as well.
Then
the general results will be applied for the Stirling's numbers of the first kind.
It seems that in this way new
results are obtained for the Stirling's numbers of the first kind.
I.
THE DEFINITION OF THE
R-NUMBERS AND THEIR
FUNDAMENTAL PROPERTIES In this paragraph we investigate numbers, which allowed us a certain generalization of the Stirling's numbers of the first kind.
-
123
-
Let G(z)
z
~
k
k=1
be an analytic function, which is regular in the neighbourhood of
z=O, where the elements of the sequence
{ak}~=1
are complex numbers. DEFINITION I. The numbers n k - , -k\ (d G (z ) ) n. . d zn z= 0
(I)
are called to be
(k = I , 2 , ... , n;
n = I , 2 , ... )
R-numbers generated by the sequence
Let n G (z) n
ak
~
ak
00
H
n
~
(z)
tions
G(z)
function
H
z=o
and n
k
z
k
k=n+1
Obviously
z
k
k=1
, k
.
is a zero of order one of the func-
G (z) n
and the zero of order
n+1
of
(z), respectively. Therefore
o
(2 )
THEOREM I. The
(k=n+l,n+2, ... ).
R-numbers,
given by Definition
depend on the elements of the sequence
PROOF. Since -
124 -
n
{ak}k=1
I,
only.
k
n
~ (k.) (~ .... dz n
j=O ] and
z=o is a zero of order k-j G (z)H (z), we have
j Gn
k-j
(Z) Hn (») Z z=O
(k-j)n+k
of the function
j
n
n
o (j=O,I, ... ,k-I),
thus dn
k
(-n Gn[(Z»)z=O dz
which is the statement of Theo em I. THEOREM 2. Let
S i
2nZn-th moment of the
M2 =(2n)!a n
f(t). We see at once that -
140 -
is
for (b)
large enough.
b
In this
section we construct representations of
the coefficients
(4) which can be handled easily when
Theorem 2 is proved.
We need the
Let
be a monotone increasing ordered
combination of order out
following notation:
of the elements
k
repetition and without permutation.
(6 )
A
a I
(al)A
a2
(a 2 ).
I, ... ,ll
with-
Let
•A
a
(a) ,
s
s
vlhe r e
A
aj
(a.,
(a.) ]
a.+I, . . . ,a.+a.-I), ] J ]
]
a. I-(a.+-a.-I) ] + ] ]
It
LS
th~t
obvious
k,
a l + ••• +a
If the
a
a
j
repr~sentation
comGination
(6)
s
+a
s
-I
holds, we say that the
is decomposed into blocks
(j= I It
(j=I, . . . ,s).
:::: 2
, ..
. , s)
with the lengths
is easy to see that the decomposition of combina-
Lions of order
k
of the elements
(I, ... ,ll)
withoUT
repetition and without permutation is unique. Returning to our duty we prove
the following state-
men t.
LEMMA
I.
ak(n) (7)
bn_k(n)
n!
(k=O, I , . . . , n- I
141
n=I,2, ... )
where 1•
(8)
(k=I ••••• n-I; n=I.2 •••• ).
and where the combination the elements
I ••••• n-I
(i I •...• i k ) of order k of without repetition and without
permutation is decomposed into blocks with lengths <X 2
a l •
····,a s · PROOF. Let n G
n
k
z .
~
(z)
k=I
According to k!
~
(
C2
1 a
n-) n z
1
a I !. .. a n ! (-1)
a)+ ..• +an=k .•• x
C
a l C <X2 3
(T)
a I +2a 2 +···+na
n
n
x ...
,
we get
i.e. the numbers
bk(n)
(k=I •••• ,n-I; n=I,2, ••• ) are
R-numbers generated by the sequence Definition 1
{e 2k - I };=I
([2J,
and Theorem 1). Thus the proof of Lemma 1
([2J, Theorem 5 and 6) is complete. (c) In this section we deal with inequalities. which will be used later. -
142 -
LEMMA 2. The sequence defined by the recursive formula
(1),
is strictly mon-
otone decreasing.
PROOF.
Since
C1
=
1
> 31 =
C 3 , we can suppose that
our Lemma is proved for the index
2k-1
(k
>
I).
If we
apply Theorem 368 of [4J in the formula (I), we obtain
which completes the proof. LEMMA 3. (k, R,=O , 1 ,2, ... ) •
PROOF. First we prove the left inequality. If
k=R,=O, then our Lemma is valid. Suppose that tru~
the Lemma is m+n
" .1l. + ~
~
n j=1
(A.-A.) ] ]
(i=I, ••• ,n-l)
j*i
and n-I A'::> A
n
n
+ n
~
j= 1
(A. -A .) , ] ]
we have
(?.8)
n IT f.(x.) ~ ~ i=I
*0
if n-I
x.E A. + n ~ ~
~
j= 1
(A. -A .) ] ]
- 206 -
(i=I, ••• ,n-l)
and n-I
:E (A. -A .) •
+
n j=1
]
]
By a theorem of Steinhaus (see [7J) the sets
A.-A. ~
~
(i=I, ••• ,n-l) tervals
contain intervals. Thus there exist in000 I. [a.,b.]CR (i=l, ••• ,n) such that
(2.9)
n
~
~
~
n
'*
f.(x.) ~
i=l
~
o
if
0,
Repeating this argument
(i=I, ••• ,.n).
x.EI. ~
~
o
(used
instead of
I.
~
A.), we ~
have n
n
(2.10)
'*
f. (x.) ~
i=l
~
0
if n-I
1
x .EI. = I~ + ~ ~ ~
o 0 :E (I.-I.) n j=1 ] ]
(i=I, ••• ,n-l)
j*i
and x
n
EI I
= I
n
O
n
n-I
+
0
0
]
]
:E (I .-I.).
n j=1
It is easy to see that the sets (2.10)
I1
(i=l, ••• ,n) in
are the intervals
(2. I 1 )
+~
n-I
+
n
1:
j=1 j*i
o
0
]
]
(a.-b.),
b~ ~
+ n
n~1 (b~-a~) 1 ] ]
j=1
j*i (i=1.2, . . . ,n-l)
-
207 -
(2. I I ) n-I ~
0
0
bO
~
~
n
(a.-b.),
n i=1
0 0]
n-I
+ ~ (b. -a .) n i=1 ~ ~
•
Thus n (2. 12)
n
f.(x.) ~ ~
i=1
'" 0,
if
(i=I, ••• ,n).
By induction, we get a sequence (i=I, •••
,n;
k=0,1,2, •••
with property
)
n-I
0
0
j= 1
]
]
:E (a .-b .), b Oo + k-
j
~
n
°1
n1 0 ~ (b.-a.) j=1
]
]
",i
(2. 13)
(i=I, .•• ,n-l)
~[
0
0
bO
~
~
k + -
n
n
'" 0,
if
Ok
n-I
n
i=1
an +
~
(a.-b.),
n-I ~
i=1
0
0
~
~
(b.-a.)
J
and n
n i=1
f
i
(x.) ~
(i=I, •••
-
208 -
,n; k=0,1,2, ••• ).
(k)
From (2.13) one can see that as
k
-
a.
_ ""
-00
~
therefore
"",
n
n
i=1 This and
f.(x.) ~ ~
(2.5)
if
'" 0,
x .ER ~
(i=I, ... ,n).
gives that
u .ER ~
(i=I, ... ,n),
which completes the proof of Lemma 2.1. Now we can easily prove
Let
THEOREM I.
XI""'X n
be continuous and
independent random variables with densities (i=I, ... ,n). Let
YI""'Y n
fi
nx
R
be continuous random var-
iables defined by the one-to-one transformation which maps the region
R -
n. y
onto the region
(2. I)
Further
suppose that the Jacobian of the inverse transformation
(2.2) exists, is continuous and does not change signs in
ny .
Then
XI'"
.,Xn
have generalized normal distribu-
tIons with densities
(B.(x.)-fJ.) ~ ~ ~
(2. 14)
f.(x.) ~ ~
fJ.ER ~
} (x.Er/ ~
=1 o
(0,
2
(x .ER'\n ~
(i=I, ... ,n)
x.
)
are arbitrary constants,
the
B; : n x.
n
R ) i f and only i f the random variables
x.~
and
onto
~
(i=I, .•. ,n)
maps the intervals
~
are independent.
-
209 -
)
~
~
functions
R
x.
PROOF.
••. ,Yn )
YI and (Y 2 , ••• then by Theorem 1.3, we have
If the random variables
are independent,
(2.15 )
g: R -
where
G : R
Rand
density functions of
YI
n-I
-
R
are the probability
(y 2 , ••• ,Yn ),
and
Hence by the help of transformation
respectively.
(2. I) we get
n g[ F I (~ i=1
( 2. I 6 )
X
XG[ F 2 (v I ' ••• , v n _ I ) , ••• , F n (v I ' ••• , v n _ I )] X
X1Fi[ i=1 ~ for all
Bi (x i ) ) ]
B.(X.)]H(VI, ••• ,v _I) ~
~
n
(xl' . . . 'x )En , where n
(i=I,2, •• • ,n-I).
x
~
~
v. = B.(x.)-B
n~
n
Furthermore
~ B~(x.)1
i=1
~~
f.(x.)
i=1
~
~
= 0,
(x )
nn
if
xERn-n x
By the substitutions
(2. 17)
Bi (xi)
(i=I, ... ,n; x .En
t.
~
~
-I
(2. 18)
(2. 19)
f. (t .) ~
~
g(z I)
fir Bi
(t i )]
I Bl[ B~I (t i g[ F I (z I )]
)]
210
)
-
,
~
(i=I, •••
I
I F i (z I ) I
-
x.
,n;
(zIER),
tiER) ,
G (z 2' ••• , Z n) =G[ F 2 (z 2' ••
0
,
Z
n) , ••
0
,
F n (z 2 ' ••• ,
Z
n)] X
(2.20)
(2.16)
(2.21)
goes over into the functional equation n n IIf.(t.)=g( ~ t.)G(tl-t , ••• ,t I-t) i=1 ~ ~ i=1 ~ n nn n ( (t 1 ' ••• , t n) ER )
for the functions
f i ,
g :
are density functions, where zero. f.
~
R -
G :
R,
Rn - I -
R.
fi,g,G
thus they cannot be almost every-
By (2.18) the same applies to the functions
(i=I,,,.,n).
Using Lemma 2.1, it follows that n
II
for all
'# 0
f.(t.)g(u 1 )G(u 2 , . " , u ) ~ ~ n
i=1
u.ER
t ., ~
(i=I, ••• ,n).
~
Now, let
be fixed and
j'#n
t.=t ~
n
if
Hj.
Then
we get from (2.21) that n f.(t.) II f.(t.)= ] ] i=1 ~ ~ Hj
=g(t.+(n-I)t ]
for all (2.22)
for all (2.23)
t.,
]
t
n
ER.
]
)G(O, . . . ,O,t.-t ]
n
,0, ... ,0)
This implies the functional equation
f . ( t . ) f (t ) n n J ] t
n
g(t.+(n-I)t ]
n
)G(t.-t ) ] n
.,t ER, where n
f(t)= n n
n IIf.(t) i=1 ~ n i'#j -
(t ER),
n
211
-
(2.24)
G'(O, .•• ,0,
G(z)o
z
(zER) •
,0, ... ,0)
j
The functional equation (2.22) is a special case of (3) with t
a=l,
b=n-I, c=l, d=-I.
Thus the functions
ER.
0' 1 , g,
f
n J n conditions of Theorem 1.1 since f
g, G
1 n (t n )¢O
Further
o
,
~
f.
are measurable. Therefore
G
g,
G
f., f
and
and
J
for all
satisfy the J
n
,
1n
are of the
:s; j
0
be two distribution e-close to each
is a constant inde-
pendent of
£.
DEFINITION 2. Let
Hand
functions. We say that tion F
is an
£-contaminated distribu-
with a contaminating function (I-£)F+£K
(i)
H
(ii)
K
interval
0
the functions
satisfy the following conditions:
-
233 -
Suppose F
(i)
F(-OO) = G(-OO) ,
(ii)
G' (x)
exists for all
x
and
1 G'
(x)
1
s
A,
00
where
J
f(t)
eitxdF(X),
-00 00
J
get)
eitxdG(x).
_00
(iii)
If(t)-g(t)
1
2. Finally we select T = A/E
I
> I.
This is possible for sufficiently small
- 235 -
E. Then
(2.12d) LT
>
2
as required by the conditions of the theorem. Since
L
is finite,
there exists an integer such
that'-'
(2. 15)
L
< (_I ) E:
n
I
The conditions of the lemma are satisfied and we see from (2.8)
that
I vex) -Sat.. (x) I It follows
2
~
c[
from (2.12c)
>
l,n 2 ,n 3 ,···,n s
convolution of
{a} n
b
some
0,
0.
gi ven by
C
n
a
r
~
and
rl,···,rs
=
n
denoting
r=O
0,
n
n
a
where
>
to be the
{C } n
Define
{b } n
and
0,0, . . . , l,n s _ 1 ,n s
~
a b
r=O r n-r
n I
n2
~
~
rl=O r2=0
5
~
r
5
=0 Consider a
! =
" " X s )'
random vector
(YI'''''Y s )
~
where
with
Xi'
Yi
for every
>
l , · · · ,n s
i=I,2, . . . ,s
°
and whenever
(X I ' ••
(i=I,2, . . . ,s)
non-negative integer-valued r.v.'s such that
= n l , •. . ,x = n ) = P s s n
=
for some P
n
>
Pn
n.
~
P(X I
=
and
°
a b
r n-r
(3. I)
P(Y
!:.)
C
n
(r.=O,I, ... ,n.; i=I,2, . . . ,s) ~
Also define
( .) X]
(j=2,3, ••• ,s)
=
(XI, . . .
and let
k=I,2, •.• ,j-1 (3.2)
P (!
i f f for some P
(3.3)
Also i f
C
r)
Po
n
Co
( 3.3)
X. ]
y(j) ]
E..I~ = !)
P (r
E..lx(j)
>
s IT i-I
denote that] (X k
> y.). Then
p (!
8 1 ,···,8 s
n
,x.), y U ) = (yl, . . . ,Y.)
x(j); and
~
>
y(j) )
(j=2,3, . . . ,s)
°
n.
e.
~
~
is true then
-
Y
and
246 -
X-Y
are independent.
PROOF. If we use the notation
P(y
=
£lx(O) = y(O)) =
P(r
x(O)=y(O)
to denote
we can see that
£)
(3.2) is
equivalent to
P(r
(3.4)
= £I~ =
P(r
=
r)
=
Elx U - 1 )
=
(Q,=1,2, ••• ,s).
Now define the sequences
V
n
s
(3.5)
b
0,0, ••. ,O,n s
P (x -
=
!:)
W
n
for fixed
s
>
r.
~
0, i=I,2, ... ,s-1
case we have that for V =V W I: n +r r n n =0 s s s s s
Q,=s
and
s
Q,=s
p
r1,···,rs_1,n s
r1,···,rs_1'0 C
r1,···,rs_1,n s
r1,···,rs_1'0
for some
r.~
n
>
s
0
O.
In this
and hence using Lemma 1 we come to
p
every
~
(3.4) is equivalent to
the conclusion that (3.4) holds for
C
n
and every
r.
~
>
0
iff n
o
s
0s
s
>
0,
(i-l,2, . . . ,s-l)
(since
were fixed but arbitrary). Consequently (3.4) for
Q,=s
holds iff p
(3.6)
C
n
n
p
n1, ••• ,ns_1'O C n1,···,ns_1'O
-
n
os s
247 -
for some
0
>
0
and
°
(i=1,2, .•• ,s). It can also be verified every n.~ > that whenever (3.6) is valid we have that, conditional on x(s-I) = y(s-I), y and x -y are independent. s s Let us now define the sequences p
(3.7)
C
r l , . . . ,r£_1 ,0, . . . ,0
(r.
~
r l , •.• ,r£_1 ,0, . . . ,0
i
and every
n£
>
fixed,
0
= 1,2, . . . ,£-1)
and
0
l:
(3.8) n
n X
'"
8£+1'"
holds for
s
=0 p(~
n £+ 1
"'£+1
for
>
···'0 s
>
.,8 s £=k,
'" s
b
o
0,
= !.)
p(x(£-I)
Assume that
£=1, . . . ,s-I.
k+I, ... ,S; 2
~
k
~
(3.4)
and is equivalent
s
to p
p
n l ,·· .,nk_l,n k , · · · ,n s
(3.9)
C
C
n l ,·· .,nk_l,n k , · · · ,n s
n I ' ••• , n k _ I ,0, .•. , 0
n X
for some
Elk""
(Note that if =y(k-I)
,8 s
(3.9)
>
0
and every
is valid then,
x
n I ' ... , n k _ I ,0, .•. ,0 8
k
n.
~
k •••
>
0
n El s
s
(i=1,2, . . . ,s).
conditional on
jk-I)=
and (Xk-Y k , Xk+I-Yk+I""'Xs-Ys) are independent.) Under these circumstances it can be shown
that,
,
y
for
£=k-I,
(3.4) is equivalent to
p
(3.10)
p
nl,···,nk_I,···,n s C
n l ,···,nk _ 2 ,O, •.. ,0 n k _ 1
C
nl,···,nk_I,···,n s
-
n l ,···,nk _ 2 ,0, ••• ,0
248 -
Elk-I
x
for some
8 k _ I ,···,8 s
... , s; 2
~
k
~
and for every
n.
~
>
0
(i=1~2,
•••
This is so because with the help of
s) •
Lemma I we can see that, for
(3.4) holds iff
R.=k-I,
p
nl,···,nk_I,O, ... ,O
(3. II)
C
nl,···,nk_I,O, ... ,O ~
(2
i.e.
k
~
s)
(by combining (3.9) and (3.11», iff (3.10) holds.
We may also observe that if (3.10) is valid then conditional on x(k-2)=y(k-2), K and (Xk_I-Y k _ 1 ,Xk-Yk , · · · .. . ,x -y ) will be independent (2 ~ k ~ s). s
s
Consequently, we can say that (3.2» (i. e.
(3.4)
(and hence
is equivalent to (3.3). Also we have that if (3.3) (3. 10) for
k=2)
holds, Y
and
X-Yare indepen-
dent. Hence Theorem 3 is established. 4. CHARACTER:ZATION OF THE MULTIPLE POISSON, BINOMIAL AND NEGATIVE BINOMIAL DISTRIBUTIONS As a result of Theorem 2 the following corollaries can be established. COROLLARY I
(Characterization of the multiple (~,!)
Poisson). Suppose that for the random vector know that s
(4. I )
P (K
~)
n i=1
n.
(~)
r. n. -r. ~
r. Pi qi
~
~
~
n.
~
i=1,2, ••• ,s)
- 249 -
~
0,
we
(i. e. multiple binomial) then condition (3.2) holds i f f n. ~ s s A. -A ~ e (i=I, ... ,s; A= ~ A.,A. > 0) P n (4.2) n ~ i= 1 ~ ~ i=1 ~ (i.e.
multiple Poisson).
PROOF. Observe that (4.1) is of the form (3. I) with n n.~ s s P. i qi ~ and b a n ~ n ~ (n ~. =0, 1 , ••• ) • n n ~ ~ i=1 i=1 s
Since the corresponding
C
n
n i=1
:n:-r
for
n.
~
~
0
the
~
Corollary follows. REMARK 2. TALWALKER [4J derived a similar characterization of the mUltiple Poisson distribution using a condition similar but more complicated than our condition (3.2). COROLLARY 2
(Characterization of the multiple
binomial). Supppose that form
P
is mUltiple Poisson of the n (4.2) and that the conditional distribution of ~1~
can be written in the form is true i f f
P(~ = £I~ =~)
(3.1).
Then condition (3.2)
is multiple binomial of the
form (4.1).
PROOF. The necessary part of the proof is straightforward and is contained in Corollary I. For "sufficiency" we observe that Theorem 2 implies that condition s n. (3.2) holds iff c = c n (A.e.) ~/n.!.Using TEICHER's n 0 i=1 ~ ~ ~ [5J extension of Raikov's theorem we see that this is so s s n. n. iff a a O n (a..) ~ In.! and b b O n (8.) ~ In.! n ~ n ~ i=1 ~ i=1 ~ {a }, {b } Since should (a. i ' 8.~ > O·, a..+8.=A.e.). ~ ~ ~ ~ n n -
250 -
satisfy the latter conditions it is immediate that we !I~
should have the distribution of
to be multiple
binomial of the form (4.1), for some COROLLARY 3
(PI, •.. ,ps)E(O,I).
(Characterization of the multiple
negative binomial). Suppose that the vector
is
such that
p (!
(4.3)
!2..)
i=1
(
-m -p i i) n. ~
(r.
$
~
n.; m.,p. ~
~
~
(i.e.
multiple negative hypergeometric).
(3.2)
holds i f f
p
n
>
0,
i=I,2, •.• ,s)
Then,
condition
is mUltiple negative binomial of
the form
(4.4)
p
(N.=m.+p.).
n
~
~
~
PROOF. The proof follows easily if one observes that (4.3) is of the form (3. I) with s
a
n
m.+n.-I
n
(~
~
ni
i=1
n.
)q.~ ~
(4.5)
s
b
n
p.+n.-I
n
(~
~
ni
i=1
s
in which case
C
n
n.
)q.~ ~
m.+p.+n.-I
n(~
i=1
~
~)
251
~
qi
ni
-
n.
-
.
REMARK 3.
It is clear that for different forms of
the sequence {a,b} characterizations for other forms n n of multivariate distributions can be obtained. Acknowledgement.
I am grateful to Dr. D.ll. SHANBHAG
for his valuable comments and helpful discussion on the subject. REFERENCES [IJ
C.R. Rao - H.
Rubin, On a characterization of the
Poisson distribution, [2J
D.N.
Sankhya A,
26(1964), 295-298.
Shanbhag, An extension of the Rao-Rubin cha-
racterization of the Poisson distribution, J. Prob.,
[3J
R.C.
Appl.
14(1977), 640-646.
Srivastava -
A.B.L.
Srivastava, On a cha-
racterization of Poisson distributions, J.
App.
Prob.,7(1970),497-501.
[4J
S. Talwalker, A characterization of the double Poisson distribution, Sankhya A, 32(1970), 265-270.
[5J
H. Teicher, On the multivariate Poisson distribution, Skand.
Aktuartidskr.,
J. Panaretos 8 Cratesicleias St. Athens 504, Greece
-
252 -
37(1954),
1-9.
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION UETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
A CHARACTERISTIC PROPERTY OF CERTAIN DISCRETE DISTRIBUTIONS J. PANARETOS -
E. XEKALAKI
INTRODUCTION
I.
A shifted univariate distribution has a probability
skG(s)
generating function (p.g.f.) of the form
G(s)
is the p.g.f. of a distribution on the integers
G(s)
0,1,2, . . . . The distribution with p.g.f. to be shifted as
where
k
is said
units to the right or left according
is a positive or negative integer.
k
In the bivariate case a shifted distribution will have p.g.f. of the form
k
m
s t G(s,t), where
G(s,t)
represents the p.g.f. of a distribution on X
{O,I, ••• }
and
k,m
{O, I , ••• } X
are integers.
Consider now two distrete random variables
x
Y. Assume that
and
Gy(s)
~
k
s Gx(s),
k
(r.v.
integer.
Then it can be shown that the factorial moments of relate to the factorial moments of r
(1. I)
L
(l)k(i)E(x(r-i))
i~O
where
z
(r)
X
z(z-I) ... (z-r+I), -
253 -
y
thus (r=0,1,2, ... )
IS)
Analogous is the expression
for the factorial mom-
ents of the vectors X = (X I 'X 2 ) and Y = (Y I ,Y 2 ) k m s t Gx(s,t) G (s,t) (k,m integers), i. e.
with
Y
( I .2)
(r=O,I,2, ••• ,
J1.=O,I,2, ••• ).
For certain types of discrete distributions the relationships between the factorial moments of their original and shifted forms reduce to expressions which can be shown to constitute a unique property. In the sequel, such properties will be used to provide characterizations for some well-known discrete univariate and bivariate distributions. Specifically, in Section 2 we provide a characterization for the geometric which subsequently is extended to characterize the class of distributions which consists of the Poisson, binomial and negative binomial distributions. A characterization of the Hermite distribution is also given. Section 3 extends the results to obtain characterizations for some bivariate distributions whose marginals are independent. Finally, Section 4 considers the case of certain bivariate dependent distributions. 2. CHARACTERIZATION OF SOME UNIVARIATE DISCRETE DISTRIBUTIONS THEOREM 2.1. Let r. v. ' s
X,Y
be non-negative discrete
such that
(2. I)
where tion
Gw(s)
denotes the p.g.f. of w. -
254 -
Then the condi-
tion
(2.2)
>
(c
is necessary and sufficient for parameter
PROOF.
c
X
r= 1 ,2 , ... )
I,
to be geometric with
-I
Necessity follows immediately.
From (1.1) we have for
Sufficiency.
k=1
(r=I,2, .•. ).
(2.3) Hence (2.2) holds if and only if (iff)
(r=I,2, ... ), i.e.
iff
o
(r=O,I,2, ... ),
which implies that (r=O, 1 ,2 , •.. ) . But this is the
r-th factorial moment about the origin
of the geometric distribution with parameter
q=c
-I
Hence the theorem is established. Note.
In the context of stochastic processes, the
characteristic property (2.2) is equivalent to the well-known lack-of-memory property (see PARZEN [3J, p.123). It has just been proved that the geometric distribution is uniquely determined by E( (x+ I )(r))
cE(x(r))
-
(r=I,2, ••.
255 -
c
>
I).
One may ask what other distributions can be characterized by similar properties. Consider for example the more general case where
c
is not a constant but
instead it is a function of
r. Specifically, let
X
be
a non-negative discrete r.v. with the property that (r=m, m+ I , ... ) for some positive integer m=1
and
C(r)=ar+b, a,b
m. Consider the simple case
>
0, i.e. (r=I,2, ... ).
(2.4)
What distributions can be characterized by this property? By the following theorem it turns out that
(2.4)
uniquely determines the class of distributions which contains precisely the Poisson, binomial and negative binomial distributions. THEOREM 2.2 (univariate case). Let Theorem 2.1.
holds i f f
X
=
(r=1 ,2, •.• ; a,b
(ar+b)E(X(r-I))
>
0)
has one of the following distributions
(i)
Poisson with parameter
( i i)
binomial with parameters for
(iii)
be as in
Then the condition
E(y(r))
(2.5)
X,Y
a
q=(a-I)/a
I.
PROOF. Necessity follows immediately. Sufficiency.
From (2.3) we have that
iff - 256 -
(2.5) holds
= 0
E(x(r»-[(a-l)r+b]E(X(r-I»
(r=I,2, •.. ),
Le. iff (2.6)
E (x (r+ I) ) _ [ (a-I) r+a+b-I] E (x (r) )
Case
o
(r=O,I,2 ..• )
a=l. Then (2.6) becomes
o
(r=O, I ,2, ••• ) •
Solving we obtain E(x(r»
= br
which implies that Case
a~l.
(r=O,I,2 •... ) X
~
Poisson (b).
We have from (2.6)
(r=O, I ,2, •.• ) .
Solving we find that (r=O, I ,2 , ••• )
(2.7)
where
z(r) = z(z+I) ... (z+r-I), z(O)=I. Obviously, for a > I, (2.6) represents the
r-th
factorial moment of the negative binomial distribution with parameters I f now
a < I
I < E(x)+1 = a+b
b = 1+ a-I we have from (3.4) for r=O
q = (a-I)/a
or
and
b
k
that
Then (2.7) becomes I-a > I.
-
257 -
I
(I_a)r(~ _I)(r)
(2.8)
I-a
o
Therefore, the distribution of i.e. =0
there exists for every
an integer
>
r
m.
:5 r :5
[~l l-aJ
-I
otherwise
denotes the integral part of
[w}
where
o
for
m
>
X
w. is terminating,
such that
0
P[ X=r} =
Then, we have from (2.6) for
E(x(m+I»-[(a-l)m+a+b-I]E(x(m»
=
r=~
0
which implies that (a-I )m+a+b-I
0
or equivalently b I-a -I
(2.9)
m
b I-a
which implies that
is a positive integer.
Hence(2.8) represents the
rth factorial moment of
the binomial distribution with parameters and
= ~ I-a
n
-I
p=l-a.
Note.
It can be seen from (2.9) that when
X
is
bounded
I~a > which (since iff
a
0)
implies
a
A
0
valid p.g.f.
Specifically, we turn our attention to the particg(s) = A1 (S-I)+A 2 (s2_ 1 ) The distribution defined by
ular case where
(2. 10)
i~
G(s)
CAL> 0, i=I,2).
i
= 1 ,2)
known in the literature as the univariate Hermite
distribution and was introduced by C.D. KEMP and A.W. - 259 -
KEMP [IJ.
It is a special case of the Poisson-binomial
distribution
(n=2)
and may be regarded as either the
distribution of the sum of two dependent Poisson variables or that of the sun of a Poisson and an independent Poisson "doublet" variable. The following theorem provides a chracteristic property for this form of generalized Poisson distribution. THEOREM 2.3. Let
X,Y
be as in Theorem 2.1.
Then
the condition
(a
holds i f f eters
X
and
a
A.W. KEMP [IJ) that if
[(x{r»
-
=
a2
PROOF. Necessity.
('.I')
0, b
I
~
•
It has been shown (C.D. KEMP and X
is Hermite
(a l ,a 2 )
then
{'a,)rl'H~[ {'a,)-l- a ('a,) --l-] +
l
(r = 0 , 1 , 2 , ••• )
where [n12] H" (x)
n
nlxn-2j
E j=O
.
(n=0,1,2, ••• ;H~(X)=I).
(n-2j)ljI2]
Moreover
(r=I,2, ••• ).
-
260 -
Combining (2.3).
[eyer»~
(2.12) and (2.13) we find that
satisfies a relationship of the form (2.11) with a b
0
and
O. Sufficiency. From (2.3) it follows that
(2.11) holds
iff
(r-I.2 .... ). i.e. iff
(r- I .2 •••• ) .
But this is the recurrence relationship that the factorial moments of the Hermite distribution with parameters -b/a
and
1/2a
Note.
The Poisson "doublet" distribution or the
(p(X-2r)=e
distribution - 0.1 •...
»
if we allow
satisfy. Hence the result.
-A
r
A /r!. P(X-2r+l) - O.
r-
can also be characterized by Theorem 2.3 b
to take on the value
O.
3. CHARACTERIZATION OF SOME BIVARIATE DISTRIBUTIONS WITH INDEPENDENT COMPONENTS We now turn to the problem of providing characterizations for bivariate versions of the distributions examined in the previous section. We first consider the simplest case of having a bivariate form with independent marginals. In what follows a bivariate distribution whose marginals are independent and of the same form will be called "double" (e.g. double Poisson). -
261 -
Indeed, by arguments which are analogous to those used in Secion 2 the following theorems can be proved to hold for double distributions. THEOREM 3. I
(characterization of the double geomet·
-
(ZI,Z2)
= (X I ,X 2 ), K = (YI'Y2)' Z =
~
ric distribution). Let
be random vectors with non-negative integer-
-valued components. Assume that
(3. I)
Then the conditions
(3.2) c
E(x(r)x(R.)
2
I
2
(r-I,2, ... ; 1=1,2, ... ; c l ,c 2 are necessary and sufficient for
X
geometric distribution with parameters
> I)
to have the double -I
cI
-I
,c 2 •
THEOREM 3.2 (characterization of the double Poisson binomial andnegative binomial distributions). Let
(X I ,X 2 ), K - (y l ,Y 2 ) and Z = (ZI,Z2) Theorem 3.1. Then the conditions
=
X
=
be as in
(3.3)
(r-I,2, •.. ; 1 ... 1,2 ••.. ; ai,b i -
262 -
> O.
i=I,2)
X
are necessary and sufficient for
to have one of the
distributions double Poisson with parameters (bl.b Z ) (i=I.Z).
(i)
a.~ =1.
(ii)
double binomial with parameters
-I+b./I-a .•
n.
~
~
(iii)
~
a.
if
~
I (i=I.Z).
An immediate consequence of Theorem 3,Z is the following theorem which enables us to characterize bivariate distributions whose marginals are not necessarily of the same form. THEOREM 3.3. Let (ZI'ZZ)
I-a. )
X.
~
for
J
= (XI.X Z ). ! = (YI.Y Z )'
be as in Theorem 3.1.
(3.3) hold i f f (i)
!
P(! =
'"
~)
~
I J
~
~
b. a. I J_I;~) (I + ___
a .-
a .
]
for
a.
~
]
b.
(-1+ ---I ~ ; -a.
(iii) X.'" binomial
J
I;
]
I-a.), X.'" neg. bin. ]
~
(i:lj;
i,j=I,Z).
THEOREM 3.4 (characterization of the double Hermite). Let X (XI.X Z ). ! = (YI.Y Z )' as in Theorem 3. I . Then the conditions
! = (ZI'ZZ) be
(3.4)
(a i
> 0, b i
0,
a~¢I, ~
i=I,Z;
bZ
b l
h)
~ )
are necessarg and sufficient for
X
to have one of
the
distributions (i)
bivariate binomial with parameters
n--h-I,
(i"I,Z), (ii) bivariate negative binomial with parameters k-h+l, PIO·(al-I)/(al+aZ-I), P OI -(a 2 -1)/(a)+a 2 -1) ai > I (i-1,2).
-
264 -
for
PROOF. Necessity follows immediately. Sufficiency. From (1.2) for
k=m=1
we have that
the conditions (4.1) hold iff
(r" I ,2, •.• ;
1- I ,2, ••• ) •
Le. iff
(4.2)
(r=O, I .2 , ••• ; 1-0, I .2 , ••• ) • The solution is given by
(4.3)
(r-0,1,2, ••• ; 1-0,1,2). a.
1 (i-I ,2). Then from (4.2) 1 < E(x.)+1 - b.+a. (i-I,2) ~
- 265 -
~
~
iff
-h
>
1.
Then (4.3) becomes
=
(4.4)
I
(I-a )I(I_a )~(_h_I)(I+~) I 2 (0 ~
o
That is when
T
below are concerned -
x'
Principal Value)
O,(PV
x"+i oo
T
! e-At(I-F(t»dt
PV
o
e f 2ni x "-ioo
>
(x"
which is obviously (x'
> A
x'+i oo
PV 2ni
f
e
0
TZ
h(z+fI) dz z
arbitrary),
arbitrary)
T(z-fl)
h(z) -;::::r::-
dz ,
x'-i oo
which, by a simple application of Cauchy's theorem on residues, is x+i oo
=
PV 2ni
f
e
T(z-A)
h(z) -;::::r::-
dz+h(A)
x-i oo (0
1) -
G
in
being precisely one
including the observation that
Tis
lower-semi-continuous. 3. SRANBAG has called attention to the related results due to MEYER [7J, p.
CROQUET and DENY, stated and proved in
152, by martingale arguments.
REFERENCES [IJ
L. Davies - R. Shimizu, On identically distributed linear statistics, Ann. Inst. Statist. Math.,
28(1976), 469-489. [2J
G. Doetsch, Introduction to the theory and application of the Laplace transformation,
translation
1974.
from the German original, Springer-Verlag,
[3J
A.M. Kagan - Ju.V. Linnik - C.R. Rao, Characterization problems in mathematical statistics,
transla-
tion from the Russian original, John Wiley,
[4J
1973.
N. Krishnaji, Note on a characterizing property of the exponential distribution, Ann. Math. Statist.,
42(1971), 361-362. [5J
Ju.V. Linnik, Linear forms and statistical criteria, I and II, Ukrainian Math.
Journal
(1953); English
translation Selected Translations in Mathematical Statistics and Probability, Amer. Math.
Providence, 8(1962).
- 291 -
Soc.,
[6]
G.
M~rsaglia
- A. Tubilla. A note on the "lack of
memory" property of the exponential distribution Ann. Prob., 3(1975). 353-354.
[7]
P.A. Meyer, Probability and potentials, Blaisdell. Waltham. Mass •• 1966.
[8]
B. Ramachandran - C.R. Rao. Solutions of functional equations arising in some regression problems. and a characterization of the Cauchy law. Sankhya A. 32(1970). 1-30.
[9]
R. Shimizu. Characteristic functions satisfying a functional equation I. Ann. Inst. Statist. Math •• 20(1968). 187-209.
[10] R. Shimizu. Solution to a functional equation and its application to some characterization problems. Research Memo. No. 131. The Institute of Statistical Mathematics. Tokyo. [ I I ] E.C. Titchmarsh,
The theory of functions, 2nd ed.,
Oxford Univ. Press. 1939. Note. The fact that if
C = G(O). our theorem need
not hold is shown by the following example due to Dr. J.S. HUANG of Guelph. Canada: take
S x S I
and
G(O) -
I-e
-I
;
G(x)
F(x)" x for -x = I-e for x
B. Ramachandran Indian Statistical Institute 7 SJSS Marg. New Delhi. 110029 India
- 292 -
0 S ~
I.
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. AUALYTIC FUNCTION l-lETHODS IN PROBABILITY THEORY DEBRECEN (HUUGARY),
1977.
ON SOME FUNDAMENTAL LEMMAS OF LINNIK B. RAHACHANDRAN
The first name that springs to mind, when one speaks of the methods of complex analysis in the theory of probability and statistics, is undoubtedly that of the late Academician YU.V. LINNIK. As is well-known, two of the major directions of his work in this area are on: (A) characterization of the normal law through the identical distribution of two linear forms in independent and identically distributed random variables (i.i.d. r.v. 's) specifically, establishing necessary and sufficient conditions for the normality of the common distribution of the latter in terms of the coefficients in the former; and (B) decomposition of probability laws. His work in area (A) was simplified by A.A. ZINGER, and an account of this simplified version may be found,
for instance, in KAGAN,
LINNIK and RAO [3J. The basic idea of using the Laplace transform to solve a certain functional equation, as well as other arguments used there, have also been found useful e1swhere, in widely different contexts: as examples. we may cite: SHIMIZU [8J,
RAMACHANDRAN and RAO [7J -
293 -
and
RAMACHANDRAN [6]. Linnik's work in area (B) was summed up in the book, LINNIK [4], and an extended and expanded
version, with simpler proof due to I.V. OSTROVSKII of many of Linnik's basic results, as also later contributions by other authors, are to be found in LINNIK and OSTROVSKII
[5].
It is the modest aim of the first section of this paper to place on record rigorized proofs of some lemmas stated by LINNIK, which are basic to this investigations in area (A): these may be found quoted as Lemma 2.4.2 in C)].
Relations
(2.4.58) and (2.4.68) there are obviously
insufficient for us to make the desired conclusions. In the second section, we state and discuss a conjecture on power-series suggested by the first of the three basic lemmas of Linnik's in his result on necessary conditions for an infinitely divisible (i.d.) law with normal component to have only i.d. components. I.
PROOF OF SOME BASIC LEMMAS OF LINNIK'S IN THE
CONTEXT OF CHARACTERIZATION OF NORMALITY THROUGH THE IDENTICAL DISTRIBUTION OF TWO LINEAR FORMS IN I. I. D.
R. V. 'S
The lemmas under reference may, as already stated, be found quoted and "proved" as Lemma 2.4.2 in C)]. We shall not consider the details of proof of all the cases to be considered, but shall only prove the following two assertions (Lemmas I. I and 1.2) as typical. LEMMA 1.1. If
[ Y1/2]
YI
is even, then
~
2
([x]
is not an even integer, while equals the largest integer
S x)
exp (-A t
2
-I t 1Y I ) - 294 -
is a ch.
f.
for all large
PROOF. Let Re z
~
O}
f(z)
=
>
A
O.
2 YI exp(-Az -z )
and 00
(I • I)
{zEC I ,
for
00
=
I e-itxf(ltl)dt
2 Ij (x)
2 ReI eitxf(t)dt. 0
_00
An obvious application of Cauchy's theorem (cf.[3J, p.
76) shows that, for any 00
( I .2)
I e
itx
f(t)dt
=
o
I e
A izx
>
0,
f(z)dz+I e
LI
izx
=
f(z)dz
II+I 2
L2
where
o s v
{z=iv
L
=
2
o
{z=u+iA
In what follows,
S A},
S u S oo}.
will denote an absolute pos-
AO
itive constant (i.e., one not depending on and
B
A
or
x)
a constant or a variable quantity which is
bounded by some absolute constant. Neither
AO
nor
B
need always denote one and the same quantity. Let us choose and fix the constants OE(O,I); A
( I .3)
>
2Y I
and
0
and
A(I+o)
A
0
x ~
0(A)
in either
and
(i)
(1.5)
decreases on
A (x)
o
S A(x)
< A for
> 0, 0 S
(I .6)
(ii) for
(1. 7)
(iii) for complex
x
e
and
(e, 00)
> I,
x -x
I
-I+x S I
2
x ,
z,
x ~ 0(A),
(a) Let us now consider the case:
A
~
AO
considered large enough; then, in view of (1.5), we have: (1.8)
(iv)
( I .9)
(v)
A[A(x)] 2 S A[A(0(A»] 2S
A(x) _< A(0(A» x 0 (A)
i
log A,
< 1 - 2A
so that (1.10)
(vi) for
0 S v S A(x),
I
I
o.
Av - IX S AA(x) - IX S
Taking
A-A(x)
in (1.2), but writing
A
or
according to convenience, we have S e -xA(x)+AA
(1.11)
S
-
B
IA
2 00
J
o
2
I -
YI
'd
-Au (u+iA), e e
exp (-A10g X+AA2) S
A 1 - (1 + I -A BA4 S x S Bx
1
fi)
- 296 -
-(YI+I)
o (x
)
u
0
and hence we have
1t
sin
2
A(x) 2 YI -vx+Av v dv YI J e A(x) -vx YI e v dv"
1t
~
sin
2 Y I 0J
=
sin
2
1t
~
0
( I. 13)
00
YI
(J e
o
-vx -Y I v dv-
00
J
-
e vXv
YI
dv).
A(x)
The second integral on the right hand side is = BX-A(I-a) for any a >
S B exp[-(I-a)xA(x)]
A
>
2y l , a
that
A(I-a)
o.
Since
may be deemed chosen sMall enough to ensure
>
(1.13) is, for
l+y l , so that the right hand side of x
~
SeA), A
~
AO'
(I • 14)
- 297 -
s
From (1.11)-(1.14), it follows that
x
~
!I(x)
>
0
for
~.AO.
S(A), A
Turning to the case
0
y(x)
0
for
0 S x S 0(A)
as well,
x2
I
212A
exp(- 4A)' proving the lemma.
LEMMA 1.2. If teger and
I
exp(- 4A(I+O)log A)
as per (1.3), and it follows from
(1.15)-(1.18) that being
A. Finally,
[Y I /2]
(HI + I)x
on choosing
I
suitably. From relations (1.19) to
H2
(1.22), we see that (b) 0 < x < 0(A)
(1.7) for (1.23)
0
y(x)
for
x
~
0(A).
: Applying (1.6) for
>
u
I
and
S I, we have
u
Y
log u -I+u Y I log u ! S B(u I log u) 2 •
Hence, for any fixed
>
a
0,
2
co
IRe f eiuX-AU {e- u
YI
Y
log u_l+u 1 10g u}du! S
o co _Au2 2Y I 2 S B feu (log u) du S
o 2 2y -a
I
(1.24)
I
-Au u J e
S B
co
du+
Setting
-(Y + I
I
'2
(I-a» +A
-(Y I +
I 2" (I +a»}
S
x
E; = 2A
Re
j
eiUX-AU2uYI
o
(1.25)
du S
I
0
S B{A
2 2y l +a
-Au u J e
2 x -4A
• Re e
co
J e
o Y
Re Ii)
+)
log u du •
-A(u-ix)2 Y I u log u du • 2
!!- E;
e -4A
J
e
A(v-0 2 Y) • 1t v (~2" +10g v)dv}
o -
30) -
+
x
+ Re {-i
For
A
2
-4A
yl+1
2
00
o
x S 0(A),
for all
Ilogl;l>][
",=AO'
Y
J e- Au (u+iO llog(u+iOdu}.
e
so that the
first term is dominated by 2 Y +1 -~ I; 2 R e { ~. 1 e 4A J e A(v-0 v Y 1 log v dv}
o
which is negative since (0,1;)
and
0, we have:
is a ch.f. i f
YI=O (mod 4); 2
Y1
(ii)
exp(-At -It I
(iii)
Y I =2(mod 4); 2 YI exp(-At -It I (a O a
ch.f.
log
2
Itl) is a ch. f.
y 1 =2(mod 4).
if
-
302 -
if
is
2. ON A BASIC LEMMA OF LINNIK'S IN THE CONTEXT OF FACTORIZING AN I.D. LAW WITH NORMAL COMPONENT - A CONJECTURE Denote by
the class of all infinitely divisible
IO
(i.d.) laws all of whose components are themselves i.d. In the course of establishing a necessary condition for an i.d.
law with normal component to belong to
I O'
Linnik enunciated three basic lemmas. The first of these three lemmas runs as follows: LEMMA. Let integers
(I
(x)
xo
0
and (ii)
for all
that
(i) PO(x)
all
n
~
>
0
Ixl >
the sets
we can establish that
x O'
x, in view of the facts respectively for all
for suitable
pq,
xo
x \I
>
and (ii)
d
n
>
0
for
O.
REFERENCES [IJ
G.D. Birkhoff algebra,
[2J
S. Maclane, A survey of modern
3rd. ed., Macmillan,
1965.
B.V. Gnedenko - A.N. Kolmogorov, Limit distributions for sums of independent random variables, 2nd ed.
Addison-Wesley, [3J
1968.
A.M. Kagan - Yu.V. Linnik - C.R. Rao, Characterization problems in mathematical statistics, John
Wiley, [4J
1973.
Yu.V. Linnik, Decomposition of probability laws (in Russian), Leningrad, Izd. Leningr. Univ.,
[5J
Yu.V. Linnik -
1960.
I.V. Ostrovskii, Decomposition of
random variables and vectors Moscow, 1972.
-
305 -
(in Russian), Nauka,
[6]
B. Ramachandran, On the strong Markov property of the exponential laws, this volume, pp.
[7]
B. Ramachandran - C.R. Rao, Solutions of functional equations arising in some regression problems, and a characterization of the Cauchy law, Sankhya A., 32 (1970), 1-30.
[8]
R. Shimizu, Characteristic functions satisfying a functional equation 20(1968),
I, Ann. Inst. Statist. Math.,
187-209.
B. Ramachandran Indian Statistical Institute 7 SJSS Marg, New Delhi,
110029 India
- 306 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ASYMPTOTIC EXPANSIONS IN A LOCAL LIMIT THEOREM* V.K. ROHATGI
1.
INTRODUCTION
Let
a sequence of independent, identically
{x } n
distributed random variables with common distribution function
and characteristic function
F
Elx 1 I m+ 2
and we suppose without restriction that
2. We shall from now on restrict attention to positive Pk
>
in (1.1). This means that we have to allow for
ak
>
I. In the next section it will become clear that we
cannot hope for more than the inf div of (1.1) for all ak
~
2.
3. THE CASE
a=2.
From (1.3) it follows that to prove the inf div of (1.1) for
ak
~
2
course, taking all
it suffices to do so for ak
equal to
2
a k =2. Of
also makes (1.1)
much more accessible for analysis. For
ak
>
2
(1.1)
cannot generally be inf div because of Lemma I.S. To see this we sketch the graph in the complex plane of the function
-
3S1 -
~
for
t
a=l
we get a half-circle).
0
in two cases: a=2
and
a
a
>
(clearly for
there exist positive
2
t l ,t 2 ,P and q=l-p i.e. there exist positive
P~a(tl )+q~a(t2)=0'
such that
2
a > 2
a "" 2
For
>
c l ,c 2 ,t o 'P and q=l-p such that P~a(cltO)+q~a(c2tO)= =0. For a=2 (or a < 2) this is not possible. We state
our findings in n
THEOREM 3 • I •
~ "-
k=1
ak
A
P k ( __ A k__ +1: )
is in general not inf
k
divif
a k >2. We now concentrate on the case
a=2,
i.e. we consid-
er (3. I)
with
Pk
>
0
(k=I,2, ... ,n)
and
0
0
nEN, A. ]
(j=I,2, ••. ,n) and
define n ~
A(z)
k=1 Let the A(z)
2n-2
zeros
z.
and
]
z. ]
(j=1,2, ••• ,n-l)
of
be ordered such that
Then m
m
~
(3.5)
ak S
k=1 for
~
Re zk
k=1
m-I,2, ••• ,n-l. REMARK. My results so far show that
arbitrary
n
and arbitrary
if m
m=1
(i*j; ~,j
I
=
I ,2) , (iii)
X. ]
'U
X.
~
negative binomial
>
I
(i*j;
(b. / (I-a. ); ~
~
(b./a.-I; ]
]
i,]=1,2).
-
373 -
(I-a
~
) / (I-a .p . » ~
(I-a.p.)/a.q.) ]]]
]
~
for
,
3. CHARACTERIZATION OF THE BIVARIATE (DEPENDENT) BINOMIAL, NEGATIVE BINOMIAL AND POISSON DISTRIBUTIONS Before proving the main result, we need to show the following
in Theorem 2.1. Assume that for some constants b./a.-I = b./a.-I = h
a.~I,
such that
~
~
E (x. 1 y
(3. I)
(Hj)
J
a.y.+(a.-I)y.+b.,
Jl)
-
~
J
~
~
~
~
J
0
< a.
(ii)
x
is bounded i f f
~
Moreover i f
(i=1,2),
0
is bounded then
X
(i-I ,2).
~
b.
~
(m l +m 2 )(I-a i )
(i=I,2), (iii)
0
< a. < ~
-I
Pi
(i=1 ,2).
PROOF. (i)
Letting
YI-Y2=0 equation
(3.1) becomes
Y S ~)
(since
o S E (x. 1 Y
-
~
(i=1,2).
b.
~
But equality cannot hold since it would imply that xI x 2
xiql q2
for all
P(~
x
x. But
= ~)/P(~
,Q)
= 0
i. e.
P (~
is non-degenerate. Hence
o
~)
b.
~
>
0
(i=1,2).
(ii)
Let
X
be bounded. Then from (3.1) since - 374 -
x
~
Y
we have m.
a.m.+m.(a.-I)+b.
~
~
]
~
~
(i:#j,
~
i,j=I,2),
i .e.
(3.2)
(i""1,2). From the positivity of
bi
it follows that
ai
