AI 中文总结
本文研究有限字母独立同分布信源的正则T-复杂度，通过结合正则恢复长度预算、理想反向链临界尺度估计等方法，证明其满足一阶熵定律，且依概率和$L^r$收敛。
AI 中文摘要
设$W_N$为固定有限字母表上严格正独立同分布信源$\boldsymbol{p}$生成的长度恰好为$N$的分组。我们证明正则T-复杂度$c_T$满足：对任意固定的$1\boldsymbol{\rho} r<\boldsymbol{\rho}$，$\frac{c_T(W_N)}{e^{-\boldsymbol{\rho}}h(\boldsymbol{p}) N/\boldsymbol{\rho} N}\boldsymbol{\rho}1$依概率且依$L^r$收敛，其中$h(\boldsymbol{p})$为以奈特为单位的信源熵，$\boldsymbol{\rho}$为欧拉-马歇罗尼常数。证明结合了正则恢复的精确长度预算、理想反向链的临界尺度$E_1$估计，以及精确的有限分组边界表示。一个精确的Doob变换恒等式将有限边界定律表示为相对于在每一步以避开当前依赖历史的后继码字为条件的理想定律。随后，历史一致的更新估计使得伸缩端点密度一致渐近于1，因此不会积累单步近似误差。
英文摘要
Let $W_N$ be an exact length-$N$ block from a strictly positive i.i.d. source $\mathbf p$ on a fixed finite alphabet. We prove that the canonical T-complexity $c_T$ satisfies \[ \frac{c_T(W_N)}{e^{-γ}h(\mathbf p) N/\log N}\longrightarrow1 \] in probability and in $L^r$ for every fixed $1\le r<\infty$, where $h(\mathbf p)$ is the source entropy in nats and $γ$ is the Euler-Mascheroni constant. The proof combines an exact length budget for canonical recovery, a critical-scale $E_1$ estimate for an ideal backward chain, and an exact finite-block boundary representation. An exact Doob-transform identity expresses the finite-boundary law relative to the ideal law conditioned at each step to avoid the current history-dependent successor codeword. A history-uniform renewal estimate then makes the telescoping endpoint density uniformly asymptotic to one, so no one-step approximation errors accumulate.
Comments10 pages, no figures
