Suppose a string X₁ ⁿ = (X_1, X₂,...,X_n) generated by a memoryless source (X_n)_{n ges 1} with distribution P is to be compressed with distortion no greater than D ges 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 X₁ ⁿ and any given codeword Y₁ ⁿ, is approximately 2^{-nR(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 ges 0 are considered; the source (X_n)_{n ges 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.

中文翻译：

---

广义渐近均分性质：充要条件

假设由分布为 P的无记忆源 (X _n ) _{n ges 1}生成的字符串 X ₁ ⁿ = (X _1, X ₂ ,...,X _n )将被压缩，失真不大于 D ges 0，使用具有分布 Q 的无记忆随机码本。压缩性能由“广义渐近均分特性”（AEP）决定，该特性表明在 X ₁ ⁿ和任何给定码字 Y ₁ ⁿ之间找到 D-close 匹配的概率为大约 2 ^{-nR(P, Q, D)} ，其中速率函数 R(P, Q, D) 可以表示为相对熵的下界。这里的主要目的是消除最近文献中出现的对该结果有效性的各种限制性假设。在抽象字母表和无界失真测度的一般设置中提供了广义AEP的充要条件。考虑所有可能的失真水平 D ges 0；源 (X _n ) _{n ges 1}可以是平稳的和遍历的；并且码本分布可以有内存。此外，匹配概率的行为被精确表征，即使广义 AEP 无效。速率函数 R(P, Q, D) 的自然特征是在同样一般的条件下建立的。