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A MULTIDIMENSIONAL LOCAL LIMIT THEOREM
FOR LATTICE DISTRIBUTIONS
by
Olaf Krafft
University of North Carolina
and
Technische Hochschule Karlsruhe
Institute of Statistics ~umeo Series No. 530
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This research was supported by the
Air Force Office of Scientific
Research Contract No,AF-AFOSR-760-65
Department of Statistics
University of North Carolina
Chapel Hill, N. C.
,t
, :_~:.
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In
.
th~s
1 l~nu·t the~rem
paper a 1oca...
., for the extreme tail of a convolution
of lattice distributed s-dimensional random vectors shall be rroved.
is applied to multinomial distributions.
The theorem is a generalization
of a result obtained by Black>vell and Hodges [1] for the one-dimensional
case and parallels a result obtained by Borovkov and Rogozin [2] for the
case of absolutely continuously distributed s-dimensional random vectors.
Let x'
= (s 1 S2' .•. 'S s)
be a random vector in the s-dimensional Eucledian
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. )
= (Kl ,K2 , ••• ,f(
A vector k'
called an integer vector if all its components K., j
J
integers, positive or negative or zero.
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I
is called a lattice with respect to a and H.
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L
b
€
a,H
= 1,
s)
€
Rs is
2, ••• ,s are
The set of all integer vectors
k is denoted by K.
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It
A set
b
: = { b
=a
+ Hk, k
€
K)
L' is a sublattice of L if
L, and L" is a proper sublattice of L if LII is a sub-
]j implies b €
lattice of L and there is at least one bEL 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 [6]:
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 LII of I,
iT
The notation
'a
such that P (x
:= ~'is used when
1
€
a
LII ) = 1.
is defined as equal to ~.
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It is shown 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 vTe "Till make the
commonly used shift from the tails to the center
d.enote the domain of convergence of the moment generating function M( z) of x,
M(Z) : = E
""
exp (z' x) P (x
= x).
N
""
x
can
,define
a class of distributions P "conjugate"
By means of M( z) one
z
to the distribution P of x, see e.g. Khinchine
~
1
P (x = x)
z
. t er10r
.
GO
sub set 0f the 1n
""
€ <I>
""
Let F be an arbitrary non-void compact
of G in Rs and
= ( b : b = grad
<D' :
For b
[5],
=M- (z) exp (z' x) p(x = x)
vThich are defined for all t € G.
In
M( z) ,
Z €
F).
""
we consider random vectors y • y(b) which take values y
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with probabilities
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may be noted that
""
(1)
where z'(b)
-1
PZ(b) (y = y) = H
€
(:!>'(b»
""
2;
=x
N
""
exp (Z.'(b)x) p(x = x)
F is that value of z for which b = gra~. (b) In M( z ).
It
(b) is that value of z for which the function
m(z,b):
attains its minimum.
= H(z:)
exp (-z'b)
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
= 1-1
(z (b»
exp (-z' (b)b),
[4], Lemma 2.
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To this end let G
2
- b
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In the following we will consider a sequence x ,x " •• of independent
l 2
random vectors with the same distribution function. x(n) ,nay be defined
n
N
TheorePL Let xl' x2 ""
.e
be a sequence of independent La, II-distributed,
random vectors with the same distribution function and let the following
conditions be satisfied:
(A)
The lattice La, II is maximal,
(B)
the determinant of the convariance matrix of xl is not
equal to zero,
(c) b
for all n,
n
the series 1-1(z + Z(b
radius and 0 as origin.
Then the following asymptotic expansion is valid uniformly for all b €~:
p-l
.
n
n
)
mn(b ) h
J + o(n- p »).
(2 ) p( ~~ x. -_
(
1
+
Z
c.(b
)nx( n ) = ',' n ,
J
j='l J n
j=l
(21tn)S/2t:.l/2(b )
I!'<J
n
Here t:.(bn) denotes the determinant of the covariance matrix 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 anYJ20sitive integer.
n
n
Proof. P (.Z x j = X(nY=~ .
.n p(Xj = x)
J=l
j .Zxj'=X(n)J=l
I!'<J
I!'<J
'....
n
)Z
' ) exp {-z I (bn"-l
- mn( b ) >
n x j ."->hj
.~~ =X
J(n
I!'<J
....
I!'<J
n(
=m
)
bn
+.),.----=-__
i. :E(i.-b
J
J
n
)=0
(i' .-b
J
n
=
I!'<J
»)IT P(Yj = x .• - b n ) =
n j=l
J
n
,A
IT P (y. =i.-b) =mn(b) P (ZY., =0).
j=l
J
J n
n
;j=l J
3
I
»:
= ~ exp{(z' + z'(b n » i)p(x l = i)
n
x
converges for all n and all z from a sphere v1ith positi ve
(D)
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I!'<J'
as x( ): = Z x . and only such values b will be of interest for which
n
. 1
J
Ja ~
-1 N
there is an x(n) such that b n : = n x (n) = b.
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i
n
~ y. = 0) we may apply the saddlepoint method used by Richer [8] •
j=l' J
....
To that end we first note that if Xl is La,H-distributed then x(n) will be
For p(
of the form na + ilk.
Hence y(b ) Hill be L H-distributed.
n
0,
(A) as it is pointed
of theorem 2 in [8] can be replaced by our condition
out by
Hecl~endorff
=~
Ey
y(b)
n
n
= M-l(Z(b )) (
n
For the expectation Ey we get
[3].
y(b) p(y
Condition (A)
= y(b
))
n
~ i l exp (z'(bn)il ) P(x l = i l ) - bn ~ exp(z'(bn)il ) P(Xl=~l))
Xl
'
Xl
=M-l(;:;(bn )) ( gradz(b ) M(z)- (gradZ(bn)'ln M(Z))M(Z(b )))
n
n
= M-l(Z(b n )) (gradz(b r(z) - gradz(b )M(z)).=
n
Since by
(1) it is seen that PZ(b ) (y =
n
p(x
1
= i
1
o.
n
y)
= 0 holds if and only if
) = 0 condition (B) implies that 6, (b )
n
=I:
0 for all b .
n
The proof for that
n
p( ~ y.
j=l J
= 0) =
p-l
h
i
(
(21tn)s/2 6,lj2(b )
n
1 + ~ c. (b ) n -j+ 0 (n -p )}
j=l J n
holds is then with slight modifications the same as it is given by
Richter [8].
The saddlepoint will be at z
o
= 0,
the expansions of the
logarithm of the moment generating function of y(b ) in Taylor series
n
will be uniform for b
n
e: 4>.
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 ~2(3!)-2
should be multiplied by 1/2 - the same correction should be made in [8],
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p. 102, line
8 f.b. - and the formula for II in the second line from the
top, p. 214 in [7], becomes
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n(K(z )
i/2";1 e
of
z~b
n
- Z
0
) and of
~(b
n
o
,.0-
,
l
f
1 + (
5
P4 (zo)
8
-:::'i."2
~'+
P
2
3
l
1
(z ,-:J )}' -n +0(-)
n
1
) might be rather cumbersome it can easily be done
for multinomial distributions:
=
Let x be an s-dimensional random vector and e , j
j
unit vectors in R.
s
= e j ) = Pj ,
p (x
The distribution of x may be given by
= 1,
j
1,2, ••• , s, be the
2, ••• ,8,
s+l
P (x = 0) = P +1' .E P. = 1, P ' > 0, j=1,2" ••• ,s+1.J
s
j=l J
x is then lattice-distributed 1vlth maximal lattice L I where I is the
0,
s-dimensional unit matrix, h = 1. The sum of n such vectors x defines
an (5 + ~)-dimensional multinomial distribution, i. e., if there are
(s + 1) pair1rlse disjoint events B. as results of an experiment with
J
P (B) = P j , j = 1, 2, ••• , s + 1, and P (n , n , .•• , n + ) is the
S l
l
2
probability that in n independent repetitions of the experiment the event
Bl occurs n l times, B occurs n2 times, ••• , BS +l occurs n s +1 times,
2
8+1
.E n,
j=l J
= n,
n. > 0, j
J -
= 1,
a, ••• ,s + 1, then
n
p(n 1 , n 2 , ••• ,n 1)= P ( .E x.
s+
j=l J
=
s
.E n. e,)
j=l J J
=
n !
n..~ n
1.
I
n·
2'··· s+l''
For the mon~nt generating function M(z) of x we get with
s S,
M( z) = P +1 + .E e J p. •
s
, 1
J
J=
-1
Let n
t
n
Nt
.E x,
-'-1
J-
J
I
= b n = (~l'~
nl
Zl
=
2""~ s) and consider for fixed
€
1'1
n
n s +1
P2 2 .. 'P s +1 •
(Sl' S2""Ss)
> 0 values
S
~,< 1 - €.
The conditions
n with~.J > €, j = 1, 2, ... ,s, ~ s +1: = 1 -.E
j=l J
of the theorem are then satisfied. The vector z(b ) can be calculated
b
n
differentiating with respect to z the function
s
s
~(z) exp (z'bn ) ~ PS+lexp (-,.E Sj ~j) + ..E p.exp(Sj(l-~j)-.E.}k~k)'
J=l
J=l J
5
k:(:,J
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The partial derivati ves will be zero for thos e values of z for 1.,hich
s
Sk
S.
~J' (PS+l + E Pke ) = p~e J, j = 1, 2, ••• ,s, i. e., if
k=l
J
~j = In (~j Ps+l / Pj
~S+l)'
This result makes it possible to calculate the functions 1/1 (z -1- z(b ))
n
and m
and
y (b ) takes values e
n
p(y
==
e. - b
J
n
=
)
= 1,
j - b n, j
f3., j
J
= 1,
2, "', s, and (-b ) with probabilities
n
2, ••. , s,
from which the covariance matrix C(bn )
C(b
.n
)
= (c ~J
.. ),c .....
~J,.,
~
=;..
= -b)
= ~S-1-1'
is calculated as
f3 ~
Hj
i j'
(1_.::1.,).::1.
""'~
""'i'
••
~=J.
Adding all remaining rows of C(b n) to the first one we get for
, 'f3 1, f32 /.. .."
- f31f32,(1-~2)f32"'"
,
.•
~(bn)
f3s
-~lf3s
- f3 l ~s' -~2 ~s,···,(l-f3s)~s
If we now add the first row, multiplied with ~j' to the j-th r01tl,
j
= 2,3, ••• ,
s, we get
s+l
A(b )
n
=
II f3~~
j=l
t.J'
With the help of (2) we can then estimate the probabilities (3) to
(4)
p(n l , n , ••• , n S +l )
2
= {(2~)s s+l
n
( n . /n) )-1/2 S+l(
/ , )n j {1+0 (-1)
1
,n nPjin
n
~,
J
J
j=l
J=l
'
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p(y
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an expression which 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.
Acknowledgement.
I am grateful to Professor W. Hoeffding for useful
discussions and constructive conunents concerning this paper.
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References
[lJ
extreme tail of a convolution.
[2J
II
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Blackwell, D. and Hodges, J. L., Jr. (1959).
multidimension~l case.
Primenen.lQ- 61-69.
App1.
[3J
Ann. Math. Statist. 30 1113-1120.
Borovkov, A. A. and Rogozin, B. A. (1965).
theorem in the
Probability in the
On the central limit
(Russian) Teor.
English translation in :theor-. Probab. and
12..
Heckendorff, H. (1965).
On a multidimensional local limit theorem
of random vectors with integer-valued coordinates.
Dop. Akad. Nauk Ukr. RSR
[4J
Verojatnos~ i
IIoeffding,
w. (
).
1. 19-23.
On probabilities of large deviations.
Fifth Berkeley Symp. on Math. Stat. and
Khinchin, A. I. (1949).
(Ulcrainian)
~rob.
To appear.
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[5J
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[6J
On the central limit theorem in Rk •
Amer. Math. Soc. 67 359-361.
[7J
Richter, W.
I.
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Statistical Mechanics.
Proc.
Dover Publications,
lIe.v York.
Ranga Rao, R. (1961).
(1957).
Local limit theorems for large deviations.
(Russian) Teor. Verojatnost. i. Primenen. 2 214-229.
translation in Theor. Probab. and App1.
[8J
Richter, W. (1958).
large deviations.
Bull.
English
2 206-220.
Multi-dimensional local limit theorems for
(Russian) Teor. Verojatnost. i Primenen.
2. 107-114. English translation in Theor. Probab. and
2. 100-106.
8
App1.