Page 184 - Elements of Distribution Theory

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6

Stochastic Processes

6.1 Introduction
Statistical methods are often applied to data exhibiting dependence; for instance, we may
observe random variables X 1 , X 2 ,... describing the properties of a system as it evolves in
time. In these cases, a model is needed for the dependence structure of the observations.
In the present chapter, we consider some probability models used for dependent data. The
usual point of view is that the X j are ordered in time, although this is not always the case.
In general, we are concerned with a collection of random variables
{X t : t ∈ T }.

For each t, X t is a random variable; although the X t may be vector-valued, here we only
consider the case in which the X t are real-valued. Such a collection of random variables is
called a stochastic process. The index set T is often either a countable set, in which case
we refer to {X t : t ∈ T } as a discrete time process,oran interval, possibly infinite, in which
case we refer to {X t : t ∈ T } as a continuous time process.
In this chapter, we consider two cases, discrete time processes in which
T = Z ≡{0, 1, 2,...}
and continuous time processes with T = [0, ∞). Note that, in both cases, the starting point
of t = 0is arbitrary and other starting points, such as t = 1, could be used if convenient;
this is sometimes done in the examples. Also, in some cases, it is mathematically convenient
to assume that the processes have an infinite history so that

T ={..., −1, 0, 1,...}
or T = (−∞, ∞). We assume that there is a set X ⊂ R such that, for each t ∈ T , the
random variable X t takes values in X; X is called the state space of the process.
Associated with each process {X t : t ∈ T } is the set of finite-dimensional distributions.
Fix a positive integer n and let t 1 ,..., t n be elements of T . Then the distribution of
)is finite-dimensional and may be handled by standard methods. The distribu-
(X t 1 ,..., X t n
tions of all such vectors for all choices of n and t 1 ,..., t n are called the finite-dimensional
distributions of the process. In general, the finite-dimensional distributions do not com-
pletely determine the distribution of the process. However, for discrete time processes it is
true that finite-dimensional distributions determine the distribution of the process. That is,
if for any n and any t 1 ,..., t n in Z,
)
(X t 1 ,..., X t n ) and (Y t 1 ,..., Y t n
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