A Central Limit Theorem for Martin-L\"of Random Numbers

We prove a Central Limit Theorem (CLT) for Martin-Löf Random (MLR) sequences. MartinLöf 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-Löf random sequence, the empirical moments and cumulative density functions (CDF) of these samples tend to their corresponding counterparts for the normal distribution.