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198 Stochastic Processes

6.19 Let {N(t): t ≥ 0} denote a nonhomogeneous Poisson process with intensity function λ(·). Let
T 1 denote the time to the first arrival. Find Pr(T 1 ≤ t) and Pr(T 1 ≤ t|N(s) = n), s > t.
6.20 Let {N(t): t ≥ 0} denote a stationary point process with N(0) = 0. Show that there exists a
constant m ≥ 0 such that E[N(t)] = mt, t > 0.
6.21 Let {N(t)! : t ≥ 0} denote a Poisson process. Find the covariance function of the process,
K(t, s) = Cov(N(t), N(s)), t ≥ 0, s ≥ 0.
6.22 Let {W t : t ≥ 0} denote a Wiener process and define
X t = Z t − tZ 1 , 0 ≤ t ≤ 1.
The process {X t :0 ≤ t ≤ 1} is known as a Brownian bridge process. Does the process {X t :
0 ≤ t ≤ 1} have independent increments?
6.23 Let {X t :0 ≤ t ≤ 1} denote a Brownian bridge process, as described in Exercise 6.22. Find the
covariance function of the process.
6.24 Let {W t : t ≥ 0} denote a Wiener process and let

X t = c(t)W f (t) , t ≥ 0
where c(·)isa continuous function and f (·)isa continuous, strictly increasing function with
f (0) = 0. Show that {X t : t ≥ 0} satisfies (W1) and (W2) and find the distribution of X t − X s ,
t > s.

6.8 Suggestions for Further Reading
The topic of stochastic processes is a vast one and this chapter gives just a brief introduction to this
field. General, mathematically rigorous, treatments of many topics in stochastic processes are given
by Cram´er and Leadbetter (1967) and Doob (1953); a more applications-oriented approach is taken
by Parzen (1962) and Ross (1995). Karlin (1975) and Karlin and Taylor (1981) provide an in-depth
treatment of a wide range of stochastic processes. Stationary and covariance-stationary processes
are discussed in detail in Yaglom (1973); see also Ash and Gardner (1975), Cox and Miller (1965,
Chapter 7), and Parzen (1962, Chapter 3).
Moving average processes are used extensively in statistical modeling; see, for example, Anderson
(1975) and Fuller (1976). Markov processes are discussed in Cox and Miller (1965); Norris (1997)
contains a detailed discussion of Markov chains. Stationary distributions of Markov chains play a
central role in the limiting behavior of the process, a topic which is beyond the scope of this book;
see, for example, Norris (1997).
Kingman (1993) gives a detailed discussion of Poisson processes; in particular, this reference
considers in detail spatial Poisson processes. Wiener processes are discussed in Billingsley (1995,
Chapter 37) and Freedman (1971).