The importance of a law of large numbers is well known. In this paper, conditional law of large numbers for Markov processes is proved, which can be used in computing quantities related to sub-Markov sequences. A variational interpretation for this limit is given, which shows that typically the quantities of interests tend to minimize a certain entropy. A quasi-compact operator argument is involved.

中文翻译：

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大数条件律和马尔可夫过程的熵

大数定律的重要性是众所周知的。本文证明了马尔可夫过程的大数条件定律，可用于计算与子马尔可夫序列有关的量。给出了对该限制的变分解释，这表明通常感兴趣的数量倾向于最小化某个熵。涉及准紧凑运算符参数。