Asymptotic equipartition property
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The asymptotic equipartition property (AEP), or Shannon-McMillan theorem, is a direct consequence of the weak law of large numbers and is used extensively in information theory. It can be summed up by the phrase 'Almost everything is almost equally probable.'
Given independent identically-distributed random variables X1, ..., Xn with entropy H(X) and p(X1, ..., Xk) the probability of observing the sequence X1, ..., Xk, the AEP states that in probability,
- <math>
{|-\frac{1}{n}} \log p(X_1, X_2, ..., X_n) - H(X)| < \epsilon \qquad \forall \epsilon>0. <math>
The AEP is used to define the typical set, which is used extensively in theories of compression.