Typical set

In information theory, the typical set is a set of sequences whose probability is close to two raised to the negative power of the entropy of their source distribution. That this set has total probability close to one is a consequence of the asymptotic equipartition property (AEP) which is a kind of law of large numbers. The notion of typicality is only concerned with the probability of a sequence and not the actual sequence itself.

This has great use in compression theory as it provides a theoretical means for compressing data, allowing us to represent any sequence Xⁿ using nH(X) bits on average, and, hence, justifying the use of entropy as a measure of information from a source.

The AEP can also be proven for a large class of stationary ergodic processes, allowing typical set to be defined in more general cases.

If a sequence x₁, ..., x_n is drawn from an i.i.d. distribution X defined over a finite alphabet ${\displaystyle {\mathcal {X}}}$, then the typical set, A_ε⁽ⁿ⁾${\displaystyle \in {\mathcal {X}}}$⁽ⁿ⁾ is defined as those sequences which satisfy:

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