Definition
Given a discrete-time stationary ergodic stochastic process on the probability space, AEP is an assertion that
where denotes the process limited to duration, and or simply denotes the entropy rate of, which must exist for all discrete-time stationary processes including the ergodic ones. AEP is proved for finite-valued (i.e. ) stationary ergodic stochastic processes in the Shannon-McMillan-Breiman theorem using the ergodic theory and for any i.i.d. sources directly using the law of large numbers in both the discrete-valued case (where is simply the entropy of a symbol) and the continuous-valued case (where is the differential entropy instead). The definition of AEP can also be extended for certain classes of continuous-time stochastic processes for which a typical set exists for long enough observation time. The convergence is proven almost sure in all cases.
Read more about this topic: Asymptotic Equipartition Property
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