..
., ..
'
ITERATED LIMITS AND THE CENTR'\L LIMIT THEOREM
FOR DEPENDENT VARIABLES
by
GEORGE MARSAGLIA
Special report to the Office of Naval Research
of work at Chapel Hill under Project NR 042 031
for resear-::h in probabilil;y and sta·:;is"Gics.
Institute of Statistics
Mime.:>g:c'uph Series No.
Febru.ary 1954
•
•
93
ITEBATED LIMITS AND THE CENTRAL LIMIT THEOREM
FOR DEPENDENT VARIABLES. 1
by
George Marsaglia
University of North Carolina
1.
Introduction.
Section 2 of this paper gives some results on
iterated limits which may be considered generalizations of well-known
results ~l, p. 254_7.
Section 3 applies these results to give easy
proofs of some central limit theorems for m-dependent variables.
2.
Iterated Probability Limits.
of an iterated limit:
lim lim a ij
j
i
=a
for constants a ij , i, j
= 1,
2, ••• ,
means
I
(1)
Here, we use the strong sense
t
lim
lim.
j ->00 \, i->Oo
I a 1j
\
- a
I I = O.
I
We note that (1) holds if, and only if, for each
E
> 0 there exist
integers M, Nl , N2, •.• such that if the pair (i,j) satisfies j >M,
1> Nj ,
< E.
then la ij - al
DEFINITION 1.
~
f,
~ij'
i, j
= 1,
-
2, ••• be random variables.
Then
plim plim f ij
j
i
means, for every E
=f
> 0,
lim lim P(lf ij - fl
j
i
1.
> e) = 0
Work sponsored by the Office of Naval Research under Contract No.
N7-onr-28402.
".
2 -
THEOREM 1.
~
~
h ij , giJ' i, j = 1, 2, •.• be random variables.
G be a function Buch that at each of its continuity points x
lim lim P(gij ~ x)
i
j
.
= G(x),
and suppose
pltm pl1m h ij
J
= O.
i
-
Then
11m lim P(giJ + h ij
J
Let
€
=
$ x) = G(x).
i
80 > 0 and a continuity point x of G be given. We shall
exhibit integers M, N , N2, ••• such that
l
(2)
if J
Ip(giJ + h ij $ x) - G(x)
>M
at x -
~,
and i
> NJ •
First, choose
~
I< €
so that G is continuous at x +
and so that
IG(x+~) .. G(x - ~)I <0.
Then choose M, Nl , N2, •.• so that, simultaneously,
,e
lhij I > 13)
<0
( 4)
p(
(5)
/P(gij$ x) - G(x)
(6)
!P(Sij ~ x .. 13) .. G(x .. 13)
I< 0
(7)
JP( Sij ~ x + 13) - G(x + 13)
I< 0
I< 0
~,
..
- 3 -
> M and i > NJ • Then for such a pair (i,J),
h ij :$ x), H(x,~) = P(giJ + hiJ S x, IhiJ 1 ~ ~),
whenever (i,J) satisfies J
let F{x)
L{x,~)
= P{giJ
+
= P{giJ ~ x, Ihij I ~~) and Q(x)
=
IF(x)-G{x) I < IF{x)-H(x.. ~) f+ fH(x .. ~ )-L(x,~)
P{giJ :$ x). We have
1+ IL(X,~ )-Q(x) 1+ IQ(x)-G{x) I.
Now by (4) and (5), each of the terms on the right except the second
is bounded by &, and since
L(x-~,~)
L(x-~,~)
-< H(x,~) -< L(X+~,~)
and
--<IH(x,~) --L(x,~) I ~ IL(X+~,~) - L(x-~,~) I < 58
L(XI~) <L(X+~,~),
by (4), (7), (3) and (6), since
/L(x+~,t3)-L(x-~,~) I $ IL(x+~,~)-Q(X+~)
1+ IQ(x+t3)-G{x+t3)(+ IG(X+~)-G(x-t3) I
+ IG{x~)-Q{x-t3) ~ IQ{x-t3)-L(x-t3,~)
Hence !F{x)-G(x) 1< 88
= E,
I
which is condition (2).
THEOREM 2. Under the conditions of Theorem 1, if there exist
constants a ij such that 11m. lim a
iJ
j
1
at x/a then 11m. 11m P{aijgij:$ x)
J i
= a > 0, .and
= G{x/a).
Using the artifice, for sUitable i, j, 7,
the proof is routine.
The details are omitted.
if
G is continuous
3. Applications to partitioned sequences of m-dependent random
variables.
Let xl' x 2' ••• be an m-dependent sequence of random variables
with zero means.
For each pair (n, k) with 2m
= x ik _k+ l
Y
i
+ ••• + x
<k
_
ik m
~
n, define
i
= 1,
2, ••.
t~ = E(~)
2
••• + x ) ,
n
Since we shall be dealing with lim lim relations, gnk' n
n
k
defined indifferently.
According to Theorems 1 and 2, if
(8)
pl1m plim h
k
(10)
then
n
g
nk
=0
lim 11m petnk ~ x) =1(21t
nk
r
- 00
< k, may be
- 5-
< x)
(11)
-
1
=-
dt.
e
(2i
The following theorems give conditions which imply (8), (9), a.nd (10).
THEOREM 3.
If there exist constants
a > 2,
(12)
B
>0
such that
n = 1, 2, ...
n = 1, 2, ...
(14)
then condition
(11)
holds.
We first esta.blish
2m
(8)
and
(9).
One readily finds, for
< k < n,
(15)
and
E(h~) ~)
(16)
(
I) ]
+
k2)&~
n
But, using
(17)
(12),
=
lim 11m
k
n
8
2
n
1- n )
= lim.. (it 11m '2
s
n
= o.
- 6 -
Relations (15), (16) and (17) imply (8) and (9).
Condition (10) will be true, by Liapounoff's Theorem
~4, p.
Now E(
284-7 if, for large k,
Iyi
ta
,)
$
ex
k
I}]
(
lim L-l
.:
n
by (14) and
ik-m
L
E(
j=1k-k+l
Ixjia ),
60
that
\
1/0
E(
Iyi
la) )
$ lim
t
nk
n
s
s
. t
n
n
nk
(8), if k is large.
THEOREM 4.
!f
xl' x , •••
2
:I.S
a stationar! Ill-dependent sequence
with zero means, then (11) holds.
For in that case, (12) holds, and, since the variances are
bounded, (8) and (9) are established as above.
(10) holds .. since, for
each k> 2m, the sequence Yl , Y2 ' .," is stationary and independent.
= O.
\.
- 7 REFERENCES
11 Cramer, H., Mathematical Methods of Statistics, Princeton, 1946.
2.
Diananda, P. H., "Some Probability L1mits with Statistical Applicacations," Froc. Camb. PhiL Soc. (1953), pp. 239-246.
3. Hoettd1ng, W. and Robbins, H. E." "The Central Limit Theorem tor
Dependent Random Variables, II Duke Mao th. Jr." (1948), pp. 773-780.
4. Uspensky" J. U., Introduction to Mathematical Probability" McGraw
Hill, 1937.