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Given an independent and identically distributed source $X = \{X_i\}_{i=1}^{\infty}$ with finite Shannon entropy or differential entropy (as thecase may be) $H(X)$, the non-asymptotic equipartition property (NEP) withrespect to $H(X)$ is established, which characterizes, for any finite blocklength $n$, how close $-{1\over n} \ln p(X_1 X_2...X_n)$ is to $H(X)$ bydetermining the information spectrum of $X_1 X_2...X_n $, i.e., thedistribution of $-{1\over n} \ln p(X_1 X_2...X_n)$. Non-asymptoticequipartition properties (with respect to conditional entropy, mutualinformation, and relative entropy) in a similar nature are also established.These non-asymptotic equipartition properties are instrumental to thedevelopment of non-asymptotic coding (including both source and channel coding)results in information theory in the same way as the asymptotic equipartitionproperty to all asymptotic coding theorems established so far in informationtheory. As an example, the NEP with respect to $H(X)$ is used to establish anon-asymptotic fixed rate source coding theorem, which reveals, for any finiteblock length $n$, a complete picture about the tradeoff between the minimumrate of fixed rate coding of $X_1...X_n$ and error probability when the errorprobability is a constant, or goes to 0 with block length $n$ at asub-polynomial, polynomial or sub-exponential speed. With the help of the NEPwith respect to other information quantities, non-asymptotic channel codingtheorems of similar nature will be established in a separate paper.

Non-asymptotic Equipartition Properties for Independent and Identically Distributed Sources