广义渐近均分性的充要条件
The Generalized Asymptotic Equipartition Property: Necessary and Sufficient Conditions
摘要：
假设由具有分布$P$的无记忆源$(X_n)_{n\geq 1}$生成的字符串$X_1^n=(X_1,X_2,...,X_n)$将使用具有分布$Q$的无记忆随机码本以不大于$D\geq 0$的失真被压缩。压缩性能由“广义渐近均分性质”（AEP）确定，AEP指出在$X_1^n$和任何给定码字$Y_1^n$之间找到$D$ -接近匹配的概率近似为$2^{-n R(P,Q,D)}$，其中速率函数$R(P,Q,D)$可以表示为相对熵的下确界。这里的主要目的是消除各种限制性假设的有效性，这一结果已经出现在最近的文献。在抽象字母表和无界失真测度的一般情况下，给出了广义AEP的充要条件。考虑所有可能的失真水平$D\geq 0$;源$(X_n)_{n\geq 1}$可以是平稳的和遍历的;并且码本分布可以具有记忆。此外，匹配概率的行为的精确特征，即使当广义AEP是无效的。率函数$R(P,Q,D)$的自然特征是在同样一般的条件下建立的。
英文摘要：
Suppose a string $X_1^n=(X_1,X_2,...,X_n)$ generated by a memoryless source $(X_n)_{n\geq 1}$ with distribution $P$ is to be compressed with distortion no greater than $D\geq 0$, using a memoryless random codebook with distribution $Q$. The compression performance is determined by the ``generalized asymptotic equipartition property'' (AEP), which states that the probability of finding a $D$-close match between $X_1^n$ and any given codeword $Y_1^n$, is approximately $2^{-n R(P,Q,D)}$, where the rate function $R(P,Q,D)$ can be expressed as an infimum of relative entropies. The main purpose here is to remove various restrictive assumptions on the validity of this result that have appeared in the recent literature. Necessary and sufficient conditions for the generalized AEP are provided in the general setting of abstract alphabets and unbounded distortion measures. All possible distortion levels $D\geq 0$ are considered; the source $(X_n)_{n\geq 1}$ can be stationary and ergodic; and the codebook distribution can have memory. Moreover, the behavior of the matching probability is precisely characterized, even when the generalized AEP is not valid. Natural characterizations of the rate function $R(P,Q,D)$ are established under equally general conditions.
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