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
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