Page 196 - Elements of Distribution Theory

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

182 Stochastic Processes

Since
∞ ∞
2
|α j+1 − α j ||α j+h+1 − α j+h |≤ (α j+1 − α j ) < ∞,
j=−∞ j=−∞
we may write
∞ ∞ ∞ ∞

R(h) = α j+1 α j+h+1 − α j α j+h+1 − α j+1 α j+h + α j α j+h
j=−∞ j=−∞ j=−∞ j=−∞
∞ ∞ ∞

= 2 α j α j+h − α j α j+h+1 − α j+1 α j+h
j=−∞ j=−∞ j=−∞
= R Z (|h|) − R Z (|h + 1|) − R Z (|h − 1|)
where R Z denotes the autocovariance function of the process {Z t : t ∈ Z}. This is in agree-
ment with Example 6.4.

6.4 Markov Processes
If X 0 , X 1 ,... are independent, real-valued random variables, then
Pr(X n+1 ≤ x|X 1 ,..., X n ) = Pr(X n+1 ≤ x)

with probability 1. If X 0 , X 1 ,... is a Markov process, this property is generalized to allow
Pr(X n+1 ≤ x|X 1 ,..., X n )to depend on X n :

Pr(X n+1 ≤ x|X 1 ,..., X n ) = Pr(X n+1 ≤ x|X n ).
That is, the conditional distribution of X n+1 given X 1 ,..., X n does not depend on X 1 ,...,
X n−1 .

Example 6.8 (First-order autoregressive process). Consider the first-order autoregressive
process considered in Example 6.2:
1

√ 2 Z 0 if t = 0
(1−ρ )
X t =
ρX t−1 + Z t if t = 1, 2,...
where Z 0 , Z 1 , Z 2 ... is a sequence of independent, identically distributed random variables,
2
each with a normal distribution with mean 0 and variance σ and −1 <ρ < 1.
Note that we may write
ρ t t−1
X t = √ Z 0 + ρ Z 1 +· · · + ρZ t−1 + Z t ;
2
(1 − ρ )
thus, for each t = 0, 1, 2,..., Z t+1 and (X 0 ,..., X t ) are independent. Using the fact that
X t+1 = ρX t + Z t+1 , t = 1, 2,...,

it follows that, for each s ∈ R,
E{exp(isX t+1 )|X 0 ,..., X t }= E{exp(isρX t )exp(isZ t+1 )|X 0 ,..., X t }
= exp(isρX t )E{exp(isZ t+1 )}.
Hence, the first-order autoregressive process is a Markov process.