We prove a Central Limit Theorem (CLT) for Martin-L\"of Random (MLR) sequences. Martin-L\"of randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make assertions about the behavior of a single random sequence, but only on the distributional behavior of a sequence of random variables. Semantically, we usually interpret CLTs as assertions about the collective behavior of infinitely many sequences. Yet, our intuition is that if a sequence of bits is "truly random", then it should provide a "source of randomness" for which CLT-type results should hold. We tackle this difficulty by using a sampling scheme that generates an infinite number of samples from a single binary sequence. We show that when we apply this scheme to a Martin-L\"of random sequence, the empirical moments and cumulative density functions (CDF) of these samples tend to their corresponding counterparts for the normal distribution.

中文翻译：

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随机数的Martin-L\"的中心极限定理

我们证明了随机 (MLR) 序列的 Martin-L\" 的中心极限定理 (CLT)。随机性的 Martin-L\" 试图捕捉比特序列“真正随机”的含义。相比之下，CLT 不对单个随机序列的行为做出断言，而仅对一系列随机变量的分布行为做出断言。在语义上，我们通常将 CLT 解释为关于无限多个序列的集体行为的断言。然而，我们的直觉是，如果一个位序列是“真正随机的”，那么它应该提供一个“随机性来源”，CLT 类型的结果应该适用于此。我们通过使用从单个二进制序列生成无限数量样本的采样方案来解决这个难题。我们表明，当我们将此方案应用于 Martin-L\"