Page 189 - Elements of Distribution Theory

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

6.3 Moving Average Processes 175

The process {X t : t ∈ Z} is known as a moving average process.Two important special
cases are the finite moving average process
q

X t = α j t− j , t = 0, 1,...,
j=0
where q is a fixed nonnegative integer, and the infinite moving average process,
∞

X t = α j t− j , t = 0, 1,....
j=0
In fact, it may be shown that a wide range of stationary processes have a representation of
the form (6.1). We will not pursue this issue here; see, for example, Doob (1953).
Before proceeding we must clarify the exact meaning of (6.1). For each n = 0, 1,...
define
n

X nt = α j t− j , t = 0, 1,....
j=−n
Then, for each t = 0, 1,..., X t is given by

X t = lim X nt . (6.2)
n→∞
Hence, we need a precise statement of the limiting operation in (6.2) and we must verify
that the limit indeed exists.
The type of limit used in this context is a limit in mean square. Let Y 0 , Y 1 ,... denote a
2
sequence of real-valued random variables such that E(Y ) < ∞, j = 0, 1,.... We say that
j
the sequence Y n , n = 0, 1,..., converges in mean square to a random variable Y if
2
lim E[(Y n − Y) ] = 0. (6.3)
n→∞
The following result gives some basic properties of this type of convergence; the proof is
left as an exercise.

Theorem 6.3. Let Y 0 , Y 1 ,... denote a sequence of real-valued random variables such that
2
E(Y ) < ∞ for all j = 0, 1,... and let Y denote a real-valued random variable such that
j
2
lim E[(Y n − Y) ] = 0.
n→∞
2
(i) E(Y ) < ∞.
(ii) Suppose Z is real-valued random variable such that
2
lim E[(Y n − Z) ] = 0.
n→∞
Then

Pr(Z = Y) = 1.
2
2
(iii) E(Y) = lim n→∞ E(Y n ) and E(Y ) = lim n→∞ E(Y ).
n
Recall that, in establishing the convergence of a sequence of real numbers, it is often
convenient to use the Cauchy criterion, which allows convergence to be established without