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 , with abbreviate notation x(t) when it does not create ambiguity, where Ω is a sample space equipped with a σ-field and a probability measure P defined on the elements of . The cumulative distribution function of x(t) is defined as (Doob 1953)
is the indicator of the set and denotes statistical expectation (ensemble average). The expected value corresponding to the distribution