Page 192 - Elements of Distribution Theory

P1: JZP
052184472Xc06 CUNY148/Severini May 24, 2005 2:41

178 Stochastic Processes

2
2
2
Furthermore, since (a + b) ≤ 2a + 2b ,
2
2
E(Y ) = E{[Y n + (Y − Y n )] } < ∞,
proving the result.
We now consider the existence of the stochastic process defined by (6.1).

Theorem 6.5. Let ..., −1 , 0 , 0 ,... denote independent random variables such that, for
each j, E( j ) = 0 and Var( j ) = 1 and let ...,α −1 ,α 0 ,α 1 ,... denote constants.
If

∞
2

α < ∞
j
j=−∞
then, for each t = 0, 1,..., the limit
n

lim α j t− j
n→∞
j=−n
exists in mean square.

Proof. Fix t = 0, 1,.... For n = 1, 2,... define
n

X nt = α j t− j .
j=−n
For m > n,
−(n+1) m

X mt − X nt = α j t− j + α j t− j
j=−m j=n+1
and
−(n+1) m

2 2 2
E[(X mt − X nt ) ] = α + α .
j
j
j=−m j=n+1
Let
n
2
A n = α , n = 1, 2,....
j
j=1
2
Since ∞ α < ∞, lim n→∞ A n exists. It follows that A n is a Cauchy sequence of real
j=−∞ j
numbers: given > 0 there exists an N 1 such that
|A n − A m | < , n, m > N 1 .
That is, given > 0, there exists an N 1 such that

m
2
α < /2, n, m > N 1 .
j
j=n+1