Page 211 - Elements of Distribution Theory

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

6.7 Exercises 197

6.12 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 .
Suppose that
1
α j = , j = 0, 1,..., q.
q + 1
Find the autocorrelation function of the process.
6.13 Let −1 , 0 , 1 ,... denote independent random variables, each with mean 0 and standard devi-
ation 1. Define
X t = α 0 t + α 1 t−1 , t = 0, 1,...
where α 0 and α 1 are constants. Is the process {X t : t ∈ Z} a Markov process?
6.14 Consider a Markov chain with state space {1, 2,..., J}.A state i is said to communicate with
a state j if

P ij (n) > 0 for some n = 0, 1,...
and
P (n) > 0 for some n = 0, 1,....
ji
Show that communication is an equivalence relation on the state space. That is, show that a
state i communicates with itself, if i communicates with j then j communicates with i, and if
i communicates with j and j communicates with k, then i communicates with k.
6.15 Let {X t : t ∈ T } denote a Markov chain with state space {1, 2,..., J}.For each t = 0, 1,...,
let Y t = (X t , X t+1 ) and consider the stochastic process {Y t : t ∈ Z}, which has state space
{1,..., J}×{1,..., J}.
Is {Y t : t ∈ T } a Markov chain?
6.16 Let Y 1 , Y 2 ,... denote independent, identically distributed random variables, such that
Pr(Y 1 = j) = p j , j = 1,..., J,
where p 1 + ··· + p j = 1. For each t = 1, 2,..., let
X t = max{Y 1 ,..., Y t }.
Is {X t : t ∈ Z} a Markov chain? If so, find the transition probability matrix.
6.17 Let P denote the transition probability matrix of a Markov chain and suppose that P is doubly
stochastic; that is, suppose that the rows and columns of P both sum to 1. Find the stationary
distribution of the Markov chain.
6.18 A counting process {N(t): t ≥ 0} is said to be a nonhomogeneous Poisson process with intensity
function λ(·)if the process satisfies (PP1) and (PP2) and, instead of (PP3), for any nonnegative
s, t, N(t + s) − N(s) has a Poisson distribution with mean
t+s
λ(u) du.
s
Assume that λ(·)isa positive, continuous function defined on [0, ∞).
Find an increasing, one-to-one function h :[0, ∞)
→ [0, ∞) such that { ˜ N(t): t ≥ 0} is a Poisson
process, where ˜ N(t) = N(h(t)).