Chapter I

Basic concepts.

Section 1.1 Stationary processes.

A (discrete-time, stochastic) process is a sequence X\, X2,..., X„,... of random

variables defined on a probability space (X, E, //,). The process has alphabet A if the

range of each X,- is contained in A. In this book the focus is onfinite-alphabetprocesses,

so, unless stated otherwise, "process" means a discrete-timefinite-alphabetprocess. Also,

unless it is clear from the context or explicitly stated stated otherwise, "measure" will

mean "probability measure" and "function" will mean "measurable function" with respect

to some appropriate a-algebra on a probability space.

The cardinality of a finite set A is denoted by \A\. The sequence am,am+\9..., ant

where each a, e A, is denoted by

anm.

The set of all such a% is denoted by AJJ,, except

for m = 1, when

An

is used.

The k-th order joint distribution of the process {X*} is the measure /x* on A* defined

by the formula

lik(a\) = Prob(X* = «*), a\ e A*.

When no confusion will result the subscript k on /x* may be omitted. The set of joint

distributions {/z*: k 1} is called the distribution of the process. The distribution of a

process can, of course, also be defined by specifying the start distribution, [i\, and the

successive conditional distributions

H{ak\a\-1) = Prob(Xt = ak\Xk^ = a*'*) = ^

M*-i(tfi )

The distribution of a process is thus a family of probability distributions, one for each

k. The sequence cannot be completely arbitrary, however, for implicit in the definition

of process is that the following consistency condition must hold for each k 1,

(1) M*(fl?) =

2M*+i(flf+1), a\eAk.

A process is considered to be defined by its joint distributions, that is, the particular

space on which the functions Xn are defined is not important; all that really matters

in probability theory is the distribution of the process. Thus one is free to choose the

underlying space (X, E, fi) on which the Xn are defined in any convenient manner, as

1

http://dx.doi.org/10.1090/gsm/013/01