Asymptotic equipartition property

The asymptotic equipartition property (AEP) 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 i.i.d random variables <math>X_1, X_2, ..., X_n</math> with entropy <math>H(X)</math> and <math>p(X_j, ..., X_k)</math> the probability of observing the sequence <math>X_j, ..., X_k</math>, the AEP states that in all probability,

<math>

{\frac{1}{n}} \log \left( \frac{1}{p(X_1, X_2, ..., X_n)} \right) \rightarrow H(X) </math>

The AEP is used to define the typical set, which is used extensively in theories of compression.