Page 186 - Elements of Distribution Theory

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

172 Stochastic Processes

Continuing in this way shows that, for any h = 0, 1,...,

D
) = (X t 1 +h ,..., X t n +h ),
(X t 1 ,..., X t n
proving the result.
Example 6.2 (First-order autoregressive process). Let Z 0 , Z 1 , Z 2 ,... denote a sequence
of independent, identically distributed random variables, each with a normal distribution
2
with mean 0 and variance σ . Let −1 <ρ < 1 and define

1
√ 2 Z 0 if t = 0
(1−ρ )
X t = .
ρX t−1 + Z t if t = 1, 2, 3,...
The stochastic process {X t : t ∈ Z} is called a first-order autoregressive process.
For each t = 0, 1, 2,..., let Y t = X t+1 . Define
√
2
˜ ρZ 0 + (1 − ρ )Z 1 if t = 0
Z t = ;
Z t+1 if t = 1, 2,...
then
1 ˜

√ 2 Z 0 if t = 0
(1−ρ )
Y t = .
˜
ρY t−1 + Z t if t = 1, 2, 3,...
˜
˜
It follows that the process {X t : t ∈ Z} is stationary provided that (Z 0 , Z 1 ,...) has the same
˜
˜
˜
˜
distribution as (Z 0 , Z 1 ,...). Clearly, Z 0 , Z 1 ,... are independent and each of Z 1 , Z 2 ,... is
normally distributed with mean 0 and standard deviation σ.It follows that {X t : t ∈ Z} has
˜
the same distribution as {Y t : t ∈ Z} and, hence, {X t : t ∈ Z} is stationary, provided that Z 0
has a normal distribution with mean 0 and standard deviation σ.
Since Z 0 and Z 1 are independent, identically distributed random variables each with
characteristic function
1 2 2

exp − σ t , −∞ < t < ∞,
2
˜
it follows that the characteristic function of Z 0 is

1 2 2 2 1 2 2 2 1 2 2
exp − σ ρ t exp − σ (1 − ρ )t = exp − σ t .
2 2 2
It follows that {X t : t ∈ Z} is stationary.
Figure 6.1 contains plots of four randomly generated first-order autoregressive processes
2
based on ρ = 0, 1/2, −1/2, 9/10 and σ = 1. Note that, in these plots, the processes are
presentedascontinuousfunctions,ratherthanaspointsatintegervaluesoft.Thesefunctions
are constructed by taking the value at an integer t to be X t and then using line segments to
connect X t and X t+1 .
The following theorem shows that certain functions of a stationary process yield another
stationary process.
Theorem 6.2. Suppose {X t : t ∈ Z} is a stationary process. For each t ∈ Z define

Y t = f (X t , X t+1 ,...)
where f is a real-valued function on X . Then {Y t : t ∈ Z} is stationary.
∞