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A MULTIDH1EHSIONAL LOCAL LIMIT THEOREM
FOR LATTICE DISTRIBUTIONS
by
Olaf Krafft
University of North Carolina
and
Technische Hochschule Karlsruhe
Institute of Statistics turneo Series No. 530
This research was supported by the
Air Force Office of Scientific
Research Contract No, AF-AFOSR-76o-65
De;)artment of Statistics
University of North Carolina
Chapel Hill, N. C.
In this paper a local limit the~rem for the extreme tail of a convolution
of lattice distributed s-dimensional random vectors shall be Droved .
. t rl'bu t'lons.
is applied to multinomial dlS
It
The theorem is a generalization
of a result obtained by Blac~iell and Hodges [lJ for the one-dimensional
case and parallels a result obtained by Borovkov and Rogo~~in [2J for the
case of absolutely continuously distributed s-dimensional random vectors.
Let x'
=
(r:.1
1 c:"2'
••• ,~';;l S ) be a random vector in the s-dimensi ::mal Eucledian
':l
space R , a' be a fixed vector in R and H be a square matrix of order
s
s with h:
s
=
1
Idet HI> 0. )
called an integer vector if all its components K., j
J
integers, positive or negative or zero.
k is denoted by K.
L
• -
a,H' -
The set of all integer vectors
A set
( b
b
=a
+
Hk, k
€
K )
is called a lattice with respect to a and H.
b
€
Ii implies b
€
= 1, 2, ••• ,s are
LJ and
L' is a sublattice of L if
L" is a proper sublattice of L if L" is a sub-
lattice of L and there is at least one b
€
L which is not in L".
A
random vector x is said to be lattice-distributed (L-distributed) if there
is a lattice L such that x is confined to L with probability one.
general this lattice L is not uniquely determined.
In
But this can be
rectified by the introduction of a concept analogous to that of the maximum
span in the one-dimensional case, see Ranga Rao [6J:
For any L-distributed random vector x we say that its lattice L is
maximal if
(i) P
(x
€
L) = 1,
(ii) there is no proper sublattice L" of L
=
il"
such that P (x € L")
1.
The nota~ion "ex :- ~tis used When ex is defined as equal to ~.
1
It is shovffi by Heckendorff [3 ] that each non-degenerate lattice-distributed
random vector has a maximal lattice.
In order to approximate the
probabilities for the extreme tail of a
convolution of lattice-distributed random vectors 'He Hill make the
cOlTImonly used shift from the tails to the center
To this end let G
denote the domain of convergence of the moment generating function M(z) of x,
,..,
N
M(z) : = ,..,E exp (z' x) P (x = x).
x
By means of M(z) one can define a class of distributions P "conjugate"
z
to the distribution P of x, see e.g. Khinchine [5],
".,
P (x
z
vThich
= x)
= M-
1
are defined for all z
subset of the interior GO
~
For b
€ ~
,..,,,,,
(z' x) p(x = x)
(z) exp
G.
€
Let F be an arbitrary non-void compact
of G in R and
s
: ={ b : b
grad
=
10
M( z) ,
z
€
F}.
,..,
we consider random vectors y • y(b) which tal;:e values y
,..,
=x
- b
with probabilities
,..,
Ifhere Z'(b)
-1
Pz (b) (y = Y ) = H
(1)
€
,..,,,,,
(z'(b)) exp (z' (b)x ) p(x = x )
F is that value of z for which b
= gra~ (b) 1n t( z ).
It
may be noted that z (b) is that value of z for which the f'unction
m(z,b): = 1,I(z) exp (-z'b)
attains its minimum.
As long as the distribution P is not concentrated
on a hyperplane in R the vector z(b) is uniquely determined by b and we
s
can define
m(b) :
see Hoeffding
= M (db))
exp (-z' (b)b),
[4], Lemma 2.
2
In the following we will consider a sequence x ,x '." of independent
l 2
random vectors with the same distribution function. x(n) :;Jay be defined
n
N
,..
as x( ): = l:
. 1
n
J~
x .
,.,
there is an
and only such values b will be of interest for ivhich
J
-1 ,..
x(n) such that b n : = n
Theorep(.. Let Xl' x2 ""
x (n) = b.
be a sequence of independent La,H-distributed
random vectors with the same distribution function and let the following
conditions be satisfied:
(A)
The lattice La, H is maximal,
(D)
the determinant of the convariance matrix of Xl is not
equal to zero,
(c)
b
(D)
Z(b )): = ~ exp{(z' + z'(b )) ;c'}p(x =;c')
l
n
n
x
converges for all n and all z from a sphere vlith positive
for all n,
n
the series H(z
-I-
radius and 0 as origin.
is valid uniformly for all b € W:
p-l
.
n
( 1 + l: C.(b )n- J + o(n-p)}~
.
J n
J='l
Here 6(bn) denotes the
deterr~nant
of the covariance rnatrix of y = y(b ),
n
the coefficients c.(b ) are dependent only on the moments of y up to the
J
n
order (2j + 2) and p is any :positive integer.
n
Proof.
n(
=m
bn
n
P (l: x· = X(n ~)=~ ,...,..
n p(Xj = ;('j) =
j=l J
j :Dcj:x(n)j=l
n
n
exp (-z' (b)l: (;c' . -b )}IT p(y. = ;c'. - b ) =
nj=l
J n j=l
J
j
n
r
__
) ~>_-=~. :r:(i .-b ) =0
J
J n
.n
IT r (y. = i.-b)
J
J n
j=l
3
= mn(b n )
n
P ( l: Y., = 0) .
~_,
,j-"-
0
n
= 0)
For p( L. Y..
-'-1'
J-
we may apply the saddlepoint method used by Richer [8].
J
To that end we first note that if xl is La,H-distributed then i(n) will be
of the form na + Hk.
Hence y(b ) Hill be L H-distributed.
n
(A) as it is pointed
of theorem 2 in [8] can be replaced by our condition
out by Heclcendorff [3].
=~
Ey
y(b)
y(b)
p(y
n
n
= M-l(Z(b n )) (
Condition (A)
0,
For the expectation Ey we get
= y(b ))
n
~
Xl
i l exp (Z'(bn);:c'l) p(x l = ;:{'l) - b
n
'
~
eXP(z'(bn);:{'l)
P(Xl=~l))
Xl
=M-l(:,(b n )) ( gradZ(b ) M(z)- (gradZ(b ) In M(z))M(Z(b )))
n
n
n
= M-l(d bn )) ( gradZ(b ill(Z) - grad i(b )M(z)) . =
n
Since by
P(x
1
=
(1) it is seen that PZ(b ) (y =
il)
o.
n
y)
= 0 holds if and only if
n
= 0 condition (B) implies that f). (b ) :I: 0 for an bn'
n
The proof for that
p-l
p
j
P(j:lYj = 0) = (2nn)S/2 f).1/2(b ) ( 1 +j:1Cj(bn ) n- + 0 (n- )}
n
h
n
holds is then with slight modifications the same as it is given by
Richter [8].
The saddlepoint '!ill be at z
= 0,
o
the expansions of the
logarithm of the moment generating function of y(b ) in Taylor series
n
will be uniform for b
n
€~.
It should be mentioned here that the
coefficients of the expansions in Richter's papers [7] and [8] are
partially incorrect.
So in formula (11) in [7] the factor P7,2(3!)-2
~
should be multiplied by 1/2 - the same correction should be made in [8],
p. 102, line 8 f.b. - and the formula for 1
top, p. 214 in [7], becomes
4
1
in the second line from the
n(K(z ) - z
i/2;(1 e
0 TO"
0
vJhereas in applications of this theorem to special cases the evaluation
of z,(u ) and of 6(b ) might be rather cumbersome it can easily be done
n
n
for multinomial distributions:
Let :: be an s-dimensional randon vector and e , j
j
unit vectors in R.
s
p (x = e.) = p., j
:J
J
1,2, ... , s, be the
=
The distribution of x may be given by
= 1,
s+l
2, ••• ,s,
P (x =0) =p
, I:
s+ 1 ,J=
. 1
P. =1, PJ' >0, j=1,2" ... ,s+1 ..
,J
x is then lattice-distributed 1vlth maximal lattice L I where I is the
0,
s-dinensional unit matrix, h = 1. The sum of n such vectors x defines
an (8 + I)-dimensional multinomia.l distribution, i. e., if there are
(s + 1) pair,rise disjoint event~ B. as results of an experiD~nt vdth
P (B.)
= p.,
J
J
J
= 1, 2, •.• ,
s + 1, and p. (n , n , .•• , n s+ 1) is the
l
2
probability that in n independent repetitions of the experiment the event
B
l
occurs n
s+1
I: n.
j=l J
l
j
times, B occurs n
2
= n, n. >
J -
0, j
= 1,
times, ••• , B + occurs n + times,
s l
S l
2
a, ••• ,s + 1, then
s
n
p(n 1 , n , ••• ,n 1)= P ( E.,.,
2
s+
j=l J
=
E
n. e.)
j=l
J J
n
n :
=-----__
n..:
1
n : ... n .-:
2
3+1
Pl
l
n
P2
2
n.
For the moment generating function H(z) of x we get ''lith z' = (Sl' S2""SS)
M(z)
-1
Let n
t
b n with
n
I:
= Ps+l
s Sj
p.
+ .Ee
J
.1=1
•
Nt
x. = b n = (~l'~ 2""~ s) and consider for fixed
~-l J
J~j > E,
€
s
j =
1, 2, ••• ,s,
~s+l: =
of the theorem are then satisfied.
1 - E
j=l
~.
< 1 -
E.
>
° values
The conditions
J
The vector z(b ) can be calculated
n
differentiating with respect to z the function
s
M(z) exp (Z'bn )
=-=
ps+lexp ( - I:
j=l
s
S. ~ .. )
J
J
5
+ E p.exp(!:.(l-I3)-Eh13k)'
j=l J
,J
"
k:J:,r
s·:- .L
••• p s+ l '
The partial deri vati ves will be zero for thos e values of z for Hhich
s
Sk
S.
~. (PS+l -I- ~ Pke ) = p~e J, j = 1, 2, ••• ,s, i. e., if
J
k=l
J
Sj = In
(~j PS-l- l / P j
~S+l)'
This result makes i t possible to calculate the functions II (z + ::,(b »
n
and
ill
(b ) to
n
P S +l
S.
s
H (z + Z(bn » = ~ (~s+l + ~ ~.e J)
~s+l
j=l J
s+l
ill
(bn )
=
II
j=l
and
~.
(PJ'/~.j) J .'
Y (b ) takes values e. - b , j = 1, 2, ••• , s, and (-b ) Hith probabilities
n
J
n
n
p(y
= = e.J
- b
n
)
~., j
J
=
= 1,
p(y
2, .•. , s,
from 'ilhich the covariance matrix C(b)
-b)
= ~S+l'
is calculated as
n
~
=
f3 f3
C(b n ) = (cij),C ij.•. = -(ll_·A j.')
~
1
H:j
f3.,
i=j.
1
Adding all remaining rows of C(b n) to the first one we get for 6(b )
n
. f3 1, f32' .. .• , f3s
- f3 1f32 ,(1-f32 )(32"'"
6(b )
n
-(31(3s
= ~s+l
A
- (31 (3s' -(32 (3s,···,(l-f3 s )f3 s
If we now add the first row, multiplied with f3., to the j-th row,
J
j = 2,3, ... , s, we get
s+1
=
II f3.~
j=l
<J'
With the help of (2) we can then estimate the probabilities (3) to
s+1
-1/2 s+1
,n.
-1
(4)
p(n , n , ... , n + ) = {(21ffi)s II (n./n»)
II (n'J .. /n.) J{l+O{n )},
1
S l
2
j=l
J
j =1
,J J
6
an expression vmich is obtained also by Hoeffding [4], applying Sterling's
formula to (3).
The function m(b) is closely related to the function
I (A (s), F) used by Hoeffding to derive more general results on probabilities of large deviations.
Acknm·Tledgement.
I am grateful to Professor W. Hoeffding for useful
discussions and constructive comments concerning this paper.
7
References
[lJ
Blackwell, D. and Hodges, J. L., Jr. (1959).
extreme tail of a convolution.
[2J
Ann. Math. Statist. 30 1113-1120.
Borovkov, A. A. and Rogozin, B. A. (1965).
theorem in the multidimensional case.
Primenen. 10 61-69.
AppL
[3J
Probability in the
On the central limit
(Russian) Teor.
English translation in Theor. Probab. and
10.
Heckendorff, H. (1965).
On a multidimensional local limit theorem
of random vectors with integer-valued coordinates.
Dop. Akad. Nauk Ukr. RSR 1
[4J
Hoeffding, W. (
).
Khinchin, A. 1.(1949).
(Ulcrainian)
19-23.
On probabilities of large deviations.
Fifth Berkeley Symp. on Math. Stat. and
[5J
Verojatnos~ i
~rob.
Statistical Mechanics.
Proc.
To appear.
Dover Publications,
rTe.. ., York.
[6J
Ranga Rao, R. (1961).
On the central limit theorem in Rl~.
BulL
Amer. ~~th. Soc. 67 359-361.
[7J
Richter, W.
(1957).
Local limit theorems for large deviations.
(Russian) Xeor. Verojatnost. i. Primenen. g 214-229.
translation in Theor. Probab. and Appl.
[8J
Richter, W. (1958).
large deviations.
l
107-114.
l
100-106.
2
English
206-220.
Multi-dimensional local limit theorems for
(Russian) Teor. Verojatnost. i Primenen.
English translation in Theor. Probab. and Anpl.
8