The autocorrelation of an ergodic process is sometimes defined as or equated to
2.
In an ergodic process, the state of the process after a long time is nearly independent of its initial state.
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These definitions have the advantage that they give sensible well-defined single-parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes.
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An important class of such continuous-time stationary process is the bandlimited stationary ergodic process with the sample space being a subset of the continuous \ mathcal { L } _ 2 functions.
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The difficulty is commonly avoided by the process which is named ergodic process, that changes with time and probability gets involved with it, and it's not always periodic at all portions of time.
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The drift-plus-penalty algorithm is known to ensure similar performance guarantees for more general ergodic processes \ omega ( t ), so that the i . i . d . assumption is not crucial to the analysis.
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AEP holds if the process is white, in which case the time samples are i . i . d ., or there exists " T " > 1 / 2 " W ", where " W " is the nominal bandwidth, such that the " T "-spaced time samples take values in a finite set, in which case we have the discrete-time finite-valued stationary ergodic process.
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