Page 191 - Elements of Distribution Theory

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

6.3 Moving Average Processes 177

It follows that
 1


2 2
m
1
2
E T 2 = E  | 
m |Y n j +1 − Y n j
j=1
m 1
2

≤ E |Y n j +1 − Y n j | 2
j=1
m
1
≤ ≤ 1.
2 j
j=1
Since, by Fatou’s lemma,
2
2
E(T ) ≤ lim inf E(T ),
m
m→∞
2
it follows that E(T ) ≤ 1 and, hence, that Pr(T < ∞) = 1. This implies that
m

Y n j +1 − Y n j
j=1
converges absolutely with probability 1.
Hence, the limit of
m

+
Y n 1 Y n j +1 − Y n j
j=1
as m →∞ exists with probability 1 so that we may define a random variable
∞

+ .
Y = Y n 1 Y n j+1 − Y n j
j=1
Note that
with probability 1.
Y = lim Y n j
j→∞
2
Consider E[(Y n − Y) ]. Since Y n j → Y as j →∞,
− Y n )
Y − Y n = lim (Y n j
j→∞
and, by Fatou’s lemma,

2 2 2
− Y n ≤ lim inf E − Y n .
E[(Y n − Y) ] = E lim Y n j Y n j
j→∞ j→∞
Fix > 0. Since Y 0 , Y 1 ,... is Cauchy in mean square, for sufficiently large n, j,

2
E Y n − Y n j ≤ .
Hence, for sufficiently large n,
2
E[(Y n − Y) ] ≤ .
Since > 0is arbitrary, it follows that
2
lim E[(Y n − Y) ] = 0.
n→∞