Page 188 - Elements of Distribution Theory

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

174 Stochastic Processes

The process is said to be second-order stationary,or covariance stationary,if µ t+h and
K(s + h, t + h)do not depend on h. Hence, {X t : t ∈ Z} is covariance stationary provided
that µ t is constant, that is, does not depend on t, and K(s, t) depends on s, t only through
the difference |s − t|;in this case we write µ t = µ and K(s, t) ≡ R(|s − t|) for some
function R on Z. The function R is known as the autocovariance function of the process;
the autocorrelation function of the process is also useful and is given by

ρ(t) = R(t)/R(0), t = 0, 1,....

Example 6.4 (Moving differences). Let {Z t : t ∈ Z} denote a covariance-stationary pro-
cess and define
X t = Z t+1 − Z t , t = 0, 1,....

According to Example 6.3, {X t : t ∈ Z} is stationary provided that {Z t : t ∈ Z} is stationary;
here we assume only that {Z t : t ∈ Z} is covariance stationary.
Clearly, E(X t ) = 0 for all t. Let Cov(Z t , Z s ) = R Z (|t − s|); then
Cov(X t , X t+h ) = 2R Z (|h|) − R Z (|h − 1|) − R Z (|h + 1|).

Since |h − 1|=| − h + 1|,it follows that Cov(X t , X t+h ) depends on h only through |h| so
that {X t : t ∈ Z} is covariance stationary.

Example 6.5 (Partial sums). Let Z 0 , Z 1 , Z 2 ,... denote a sequence of independent, iden-
tically distributed random variables, each with mean 0 and standard deviation σ. Consider
the process defined by
X t = Z 0 + ··· + Z t , t = 0, 1,....

2
Then E(X t ) = 0 and Var(X t ) = tσ ; hence, the process is not stationary.
The variance of the process can be stabilized by considering
Z 0 +· · · + Z t
Y t = √ , t = 1, 2,...
(t + 1)
2
which satisfies E(Y t ) = 0 and Var(Y t ) = σ for all t = 0, 1, 2,.... However,
σ 2
Cov(Y t , Y s ) = √ min(s + 1, t + 1)
[(s + 1)(t + 1)]
so that {Y t : t ∈ Z} is not covariance stationary.

6.3 Moving Average Processes
Let ..., −1 , 0 , 1 ,... denote a doubly infinite sequence of independent random variables
such that, for each j,E( j ) = 0 and Var( j ) = 1 and let ...,α −1 ,α 0 ,α 1 ,... denote a doubly
infinite sequence of constants. Consider the stochastic process {X t : t ∈ Z} defined by

∞

X t = α j t− j , t = 0, 1,.... (6.1)
j=−∞