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In information theory, the asymptotic equipartition property (AEP) is a general property of the output samples of a stochastic source. It is fundamental to the concept of ''typical set'' used in theories of compression. Roughly speaking, the theorem states that although there are many series of results that may be produced by a random process, the one actually produced is most probably from a loosely defined set of outcomes that all have approximately the same chance of being the one actually realized. (This is a consequence of the law of large numbers and ergodic theory.) Although there are individual outcomes which have a higher probability than any outcome in this set, the vast number of outcomes in the set almost guarantees that the outcome will come from the set. One way of intuitively understanding the property is through Cramér's large deviation theorem, which states that the probability of a large deviation from mean decays exponentially with the number of samples. Such results are studied in large deviations theory; intuitively, it is the large deviations that would violate equipartition, but these are unlikely. In the field of pseudorandom number generation, a candidate generator of undetermined quality whose output sequence lies too far outside the typical set by some statistical criteria is rejected as insufficiently random. Thus, although the typical set is loosely defined, practical notions arise concerning ''sufficient'' typicality. == Definition == Given a discrete-time stationary ergodic stochastic process ''X'' on the probability space (Ω, ''B'', ''p''), AEP is an assertion that : where ''H''(''X'') or simply ''H'' denotes the entropy rate of ''X'', which must exist for all discrete-time stationary processes including the ergodic ones. AEP is proved for finite-valued (i.e. |Ω| < ∞) 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 ''H'' is simply the entropy of a symbol) and the continuous-valued case (where ''H'' 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Asymptotic equipartition property」の詳細全文を読む スポンサード リンク
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