T-spaced time samples take values in a finite set, in which case we have the discrete-time finite-valued stationary ergodic process. Any time-invariant operations also preserves the asymptotic equipartition property, stationarity and ergodicity and we may easily turn a stationary process to non-stationary without losing the asymptotic equipartition property by nulling out a finite number of time samples in the process. Category theory A category theoretic definition for the equipartition property is given by Gromov. Given a sequence of Cartesian powers of a measure space P, this sequence admits an asymptotically equivalent sequence HN of homogeneous measure spaces (i.e. all sets have the same measure; all morphisms are invariant under the group of automorphisms, and thus factor as a morphism to the terminal object) . The above requires a definition of asymptotic equivalence. This is given in terms of a distance function, giving how much an injective correspondence differs from an isomorphism. An injective correspondence is a partially defined map that is a bijection; that is, it is a bijection between a subset and . Then define where |S| denotes the measure of a set S. In what follows, the measure of P and Q are taken to be 1, so that the measure spaces are probability spaces. This distance is commonly known as the earth mover's distance or Wasserstein metric. Similarly, define with taken to be the counting measure on P. Thus, this definition requires that P be a finite measure space. Finally, let A sequence of injective correspondences are then asymptotically equivalent when Given a homogenous space sequence HN that is asymptotically equivalent to PN, the entropy H(P) of P may be taken as See also Cramér's theorem (large deviations) Shannon's source coding theorem Noisy-channel coding theorem Notes References Journal articles Claude E. Shannon. "A Mathematical Theory of Communication". Bell System Technical Journal, July/October 1948. Sergio Verdu and Te Sun Han. "The Role of the Asymptotic Equipartition Property in Noiseless Source Coding." IEEE Transactions on Information Theory, 43(3): 847–857, 1997. Textbooks Information theory Theorems in statistics