Theory of Statistics - George Mason University

122 1 Probability Theory
States, Times, Notation, and Basic Definitions
The smallest set of measure 1 is called the state space of a stochastic process;
that is, the range of X is called the state space. Any point in the state space
is called a state.
If the index set of a stochastic process is countable, we say the process
is a discrete time stochastic process. We can index a discrete time process
by 0, 1, 2, . . ., especially if there is a fixed starting point, although often
. . ., −2, −1, 0, 1, 2, .. . is more appropriate.
In many applications, however, the index of a stochastic process ranges
over a continuous interval. In that case, we often use a slightly different notation
for the index set. We often consider the index set to be the interval
[0, T], which of course could be transformed into any finite closed interval. If
the index set is a real interval we say the process is a continuous time stochastic
process. For continuous time stochastic processes, we sometimes use the
notation X(t), although we also use Xt. We will discuss continuous time processes
in Section 1.6.2 below and consider a simple continuous time process
in Example 1.32.
A property that seems to occur often in applications and, when it does, affords
considerable simplifications for analyses is the conditional independence
of the future on the past given the present. This property, called the Markov
property, can be made precise.
Definition 1.50 (Markov property)
Suppose in the sequence {Xt}, for any set t0 < t1 < · · · < tn < t and any x,
we have
Pr(Xt ≤ x | Xt0, Xt1, . . ., Xtn) = Pr(Xt ≤ x | Xtn). (1.258)
Then {Xt} is said to be a Markov sequence or the sequence is said to be
Markovian. The condition expressed in equation (1.258) is called the Markov
property.
Definition 1.51 (homogeneous process)
If the marginal distribution of X(t) is independent of t, the process is said to
be homogeneous.
***fix Many concepts are more easily defined for discrete time processes,
although most have analogs for continuous time processes.
Definition 1.52 (stopping time)
Given a discrete time stochastic process ***fix change to continuous time
a random variable
X : {0, 1, 2, . ..} × Ω ↦→ IR,
T : Ω ↦→ {0, 1, 2, . . .} (1.259)
is called a stopping time if the event {T = t} depends only on X0, . . ., Xt for
n = 0, 1, 2, . ...
Theory of Statistics c○2000–2013 James E. Gentle