  We prove two theorems related to the 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. We also prove the well known almost sure central limit theorem
(ASCLT), which provides an alternative, albeit less intuitive, answer to this
question. Both results are also generalized for Schnorr random sequences.
