Page 209 - Elements of Distribution Theory

P1: JZP
052184472Xc06 CUNY148/Severini May 24, 2005 2:41

6.7 Exercises 195

. Then, since there is a one-to-one correspondence between {Z 1 , Z 2 ,..., Z n } and
Z n = W t n
{Z 1 , Z 2 − Z 1 , Z 3 − Z 2 ,..., Z n − Z n−1 },
E[Z n+1 |Z 1 ,..., Z n ] = E[Z n+1 |Z 1 , Z 2 − Z 1 ,..., Z n − Z n−1 ]
= Z n + E[Z n+1 − Z n |Z 1 , Z 2 − Z 1 ,..., Z n − Z n−1 ].
Since
E[Z n+1 − Z n |Z 1 , Z 2 − Z 1 ,..., Z n − Z n−1 ]
]
= E[Z t n+1 − Z t n |Z t 1 , Z t 2 − Z t 1 ,..., Z t n − Z t n−1
= 0
by properties (W2) and (W3), it follows that

E[Z n+1 |Z 1 ,..., Z n ] = Z n ;
that is, the process {Z t : t = 1, 2,...} is a martingale.

6.7 Exercises

6.1 For some q = 1, 2,..., let Y 1 ,..., Y q and Z 1 ,..., Z q denote real-valued random variables such
that
E(Y j ) = E(Z j ) = 0, j = 1,..., q,

2 2 2
E Y j = E Z j = σ , j = 1,..., q,
j
for some positive constants σ 1 ,...,σ q ,
E(Y i Y j ) = E(Z i Z j ) = 0 for i = j
and E(Y i Z j ) = 0 for all i, j.
Let α 1 ,...,α q denote constants and define a stochastic process {X t : t ∈ Z} by
q

X t = [Y j cos(α j t) + Z j sin(α j t)], t = 0,....
j=1
Find the mean and covariance functions of this process. Is the process covariance stationary?
6.2 Let Z −1 , Z 0 , Z 1 ,... denote independent, identically distributed real-valued random variables,
each with an absolutely continuous distribution. For each t = 0, 1,..., define

1 if Z t > Z t−1
X t = .
−1 if Z t ≤ Z t−1
Find the mean and covariance functions of the stochastic process {X t : t ∈ Z}.Is this process
covariance stationary? Is the process stationary?
6.3 Let {X t : t ∈ Z} and {Y t : t ∈ Z} denote stationary stochastic processes. Is the process {X t + Y t :
t ∈ Z} stationary?
6.4 Let Z 0 , Z 1 ,... denote a sequence of independent, identically distributed random variables; let
X t = max{Z t ,..., Z t+s }, t = 0, 1,...
where s is a fixed positive integer and let

Y t = max{Z 0 ,..., Z t }, t = 0, 1,....
Is {X t : t ∈ Z} a stationary process? Is {Y t : t ∈ Z} a stationary process?