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:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)