A CENTRAL LngIT THEOREH FOR m-DEPENDENT VARIANCES
by
P H Diananda
Institute of Statistics, University of North Cerolina
and Department of Mathematics, university of Malaya
Institute of Statistics
ili[irneograph Series No. 94
February 3, 1954
A CENTRAL LIMIT TIIEOREM FOR m-DEPENDENT VARIABLES
by
P. H. Diananda
Institute of Statistics, Chapel Hill
Smmnary.
1.
In this paper a central limit theorem is obtained
for a sequence of m-dependent random variables with bounded yth moments
(y > 2) and with the property that, for large n, n/s 2 is bounded, where
n
s
n
is the standard deviation of the nth partial sum of the sequence.
2.
Preliminaries.
In this section we explain some terms and
notations and mention two theorems which we employ.
A sequence of random variables
(i
=t
1, 2, ... )
(1)
is said to be m-dependent if the random variables (Xl' ••• , X ) and
a
(Xb , ••. , Xc) (1 ~ a, b < c) are independent whenever b - a> m •
We define
Sn
= Xl
(2)
+ ••. + X
n •
Then if the X. have zero means and finite variances
1
We write
~
. for the yth absolute moment of X..
yl
1
Thus
2
~ yi
:: E
IxrI
(4)
(i::: 1, 2, ... ) •
The two theorems which we use are the following:
1heorem 1
~.
L-Z].
~
tWn }
(n:: 1, 2, •• 0) be a sequence of random vari-
Let n ::: hk + r (h, k integers;
0:5 r < k).
Suppose that, for
(n, k), there exist random variables Un, k' Vn, k and distribution
functions Fk(x), F(x) such that
each
n =Uk+V
n,
n, k
1f.T
,
the distribution function of U k ---> Fk(x) as h --->
(b)
n,
-
00
(for all r and for k~ K) ~ x is a continuity point o£ Fk(x) ,
Fk(x) - > F(x) ~ k - >
(c)
00
g
x is a continuity point of
F(x), and
(d)
h, k - >
V k converges in probability to zero uniformly in r
n,
00
~
•
Then the distribution funCtion of W - > F(x) as n --->
.
n
-
00
if x is a
---
-
continuity point of F(x) •
Theorem 2 ~1_7.
Let (1) be a sequence of independent random variables
with means zero and finite yth absolute moments for some y > 2.
as n - >
Suppose,
00 ,
1
n
l:
i=l
,B.
yJ. - >
O.
(,)
3
Then the distribution function of Sn /s n ---> the standardized normal distribution function •
3.
Theorem 3.
The central limit theorem.
We shall prove the following.
Let (1) be a sequence of m-dependent random variables' with means
zero and yth absolute moments bounded for
so~
y > 2.
Suppose,
~
n --->
00,
Then the distribution function of Sn Is n ---> the standardized normal distribution function.
Let h, k, r be as in Theorem 1.
Define
l, ... ,h),
Vn, 1C; 1 = (Zl
V
n,k;3
+ ••• +
=
zh)/sn
'
vn,k;2 '" (Sn
1
1
(Y + ... + Y ) ( - - - )
l
h
s n s n, k
.. .'3,
nk
)/ s
n
,
,
(11)
4
V k3,1ri=U k+ V k
Vn, k=Vn,;
k1+ Vn, k2+
;
n, ;
n
n,
n,
wn = sn /s n
Then
Lemma 1.
For large fixed k, ~ h - >
(10 )
00
,
L ;:
Proof.
E
(11)
I yI I =
E
I
{X(i-l)k+l + : .• +
k-m
< (k"'m)y-1
-
Z
g=1
E
l
Y
X(i-l)k+g
xik_ml ..,
I
I
where, by data of Theorem 3,
E
I I
Xi
<
bY <
00
(13)
•
Also, from (8) ,
2
+ Zh) }
•
e·
5
From (13), since y > 2, we have that
Hence, from (7),
(16)
so that, from (14),
,
where
161
~
1.
(;{.7)
Thus, by (6),
s
2
n,k
-.,...-
(18)
=:
C.
shk
as k - >
00
uniformly in h and in r.
s
From (11), (12) and (19),
2
n,k
Hence, for large fixed k,
(19 )
6
as h--->
for k fixed (and large).
00,
(20)
->0
L<
The lemma is thus proved.
Lemma 2.
For large k, the distribution function of U k ---> the standardn,
ized normal distribution function (Fk(x) = F(x), say) as h ---> 00
If k ~ 2m (as would happen for large k) the Y
Proof.
Hence we can apply Theorem 2 making use of Lemma 1.
Lemma 3.
V
h, k --->
00 •
Proof.
We have, from (8), (6), (16), (15)
are independent.
i
This gives us Lemma 2.
k,converges in probability to zero uniformly in r _as
n, _ ----:.'----~----"'----------''---
(21)
and
unifomly in r as h, k - >
E(V2
J
n, k'3
,
)
Now, from (8) and
= s2
n, k
(3),
00.
Also
(2 .. -1:..)2
sn
sn,k
a
= (
n,k
Shk
shk
S
n
.. 1)
2
•
.
.
7
·e
shk
2
~
s
E
=:
n
S
J = E {2;
~
-Vn" k'2]
s
sn
{ Shk
2
2
n
uniformly in r as h,k --->
00.
Using (24) and (18) in (23) we get
2
E(Vn, k'3)
-> 0
,
uniformly in r as h, k - >
00.
From (21), (22), (25) and (9) the lemma
follows.
We can now complete the proof of Theorem 3 using Lemmas 2, 3 and
Theorem 1 and noting (10), the conditions of Theorem 1 being all satisfied.
References
/-1
- -
7 cramer,
H.
Random variables and probability distributions (Cambridge,
1937), P. 60.
;-2 7 Diananda, P. H. I~ome probability limit theorems with statistical
- applications,1I Froc. Camb. Phil. Soc. 49 (1953), 239-46.