Page 210 - Elements of Distribution Theory

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

196 Stochastic Processes

6.5 Let {X t : t ∈ Z} denote a stationary stochastic process and, for s, t ∈ Z, let K(s, t) =
Cov(X s , X t ). Show that if
lim K(s, t) = K(0, 0)
s,t→∞
then there exists a random variable X such that
2
lim E[(X n − X) ] = 0.
n→∞
6.6 Let Y, Y 0 , Y 1 ,... denote real-valued random variables such that
2
lim E[(Y n − Y) ] = 0.
n→∞
Let a, a 0 , a 1 ,... and b, b 0 , b 1 ,... denote constants such that
lim a n = a and lim b n = b.
n→∞ n→∞
For n = 0, 1,..., let X n = a n Y n + b n and let X = aY + b. Does it follow that
2
lim E[(X n − X) ] = 0?
n→∞
6.7 Let {X t : t ∈ Z} denote a moving average process. Define

Y t = m c j X t− j , t = 0, 1,...
j=0
for some constants c 0 , c 1 ,..., c m .Is {Y t : t ∈ Z} amoving average process?
6.8 Let {X t : t ∈ Z} denote a stationary stochastic process with autocovariance function R(·). The
autocovariance generating function of the process is defined as
∞
j
C(z) = R( j)z , |z|≤ 1.
j=−∞
Show that the autocorrelation function of the process can be obtained by differentiating C(·).
6.9 Let {X t : t ∈ Z} denote a finite moving average process of the form
q

X t = α j t− j
j=0
2
where 0 , 1 ,... are uncorrelated random variables each with mean 0 and finite variance σ . Let
C(·) denote the autocovariance generating function of the process (see Exercise 6.7) and define
q

D(z) = α j z t− j , |z|≤ 1.
j=0
Show that
2
−1
C(z) = σ D(z)D(z ), |z|≤ 1.
6.10 Prove Theorem 6.3.
6.11 Let R(·) denote the autocovariance function of a stationary stochastic process. Show that R(·)
is positive semi-definite in the sense that for all t 1 < t 2 < ... < t m , where m = 1,..., and all
real numbers z 1 , z 2 ,..., z m ,
m m
R(t i − t j )z i z j ≥ 0.
i=1 j=1