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
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)