1.1 Second-Order Characterization of Stochastic Processes

1.1.1 Time-Domain Characterization

In the classical stochastic-process framework, statistical functions are defined in terms of ensemble averages of functions of the process and its time-shifted versions. Nonstationary processes have these statistical functions that depend on time.

Let us consider a continuous-time real-valued process img, with abbreviate notation x(t) when it does not create ambiguity, where Ω is a sample space equipped with a σ-field img and a probability measure P defined on the elements of img. The cumulative distribution function of x(t) is defined as (Doob 1953)

(1.1) equation

where

(1.2) equation

is the indicator of the set img and img denotes statistical expectation (ensemble average). The expected value corresponding to the distribution

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