Page 194 - Elements of Distribution Theory

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

180 Stochastic Processes

since
∞ ∞
2 2
(α j − α j+h ) ≤ 4 α < ∞,
j
j=−∞ j=−∞
it follows that X nt − Y nt converges to X t − Y t in mean square. Hence,
∞
2
Var(X t − Y t ) = lim Var(X nt − Y nt ) = (α j − α j+h ) .
n→∞
j=−∞
Similarly,
∞

Var(X t ) = Var(Y t ) = α j ;
j=−∞
hence,
∞ ∞ ∞
2 2
2Cov(X t , Y t ) = 2Cov(X t , X t+h ) = 2 α − (α j − α j+h ) = 2 α j α j+h ,
j
j=−∞ j=−∞ j=−∞
proving the first result.
To prove the second result, let Y t = X t+1 , t = 0, 1,.... Then

Y t = α j t+1− j , t = 0, 1,....
j=−∞
Since ..., −1 , 0 , 1 ,... are identically distributed, we may write

Y t = α j t− j , t = 0, 1,....
j=−∞
It follows that the process {Y t : t ∈ Z} has the same structure as {X t : t ∈ Z}.Itnow follows
from Theorem 6.1 that {X t : t ∈ Z} is stationary.

Example 6.6 (Finite moving average process). Consider the qth-order finite moving aver-
age process,

q

X t = α j t− j , t = 0, 1,...,
j=0
where q is a fixed nonnegative integer; here −q , −q+1 ,... is a sequence of independent
random variables such that, for each j,E( j ) = 0 and Var( j ) = 1 and α 0 ,α 1 ,...,α q
are constants. This model is of the general form considered above with α i = 0, i ≤−1,
i ≥ q + 1; hence, the condition of Theorem 6.6 is satisfied.
It follows that the covariance function of the process is given by
q−h

R(h) = j=0 α j α j+h if h = 0, 1,..., q .
0 if h = q + 1, q + 2,...
Thus, observations sufficiently far apart in time are uncorrelated.
Figure 6.2 contains plots of four randomly generated moving average processes with
α j = 1, j = 1, 2,..., q, and with the j taken to be standard normal random variables, for
q = 0, 1, 2, 5. As in Figure 6.1, the processes are shown as continuous functions, rather
than as points.