We present a proof of a martingale central limit theorem (Theorem 2) due to McLeish (1974). Then, an application to Markov chains is given.

Now we recall the statement of the central limit theorem (CLT) and give a proof in the case of IID (independent identically distributed) random variables. The weak law of large numbers says that if Xn is a sequence of IID random variables with E[Xn] = 0, then writing Sn = ∑n−1 k=0 Xk, the time averages 1 n Sn converge to 0 in probability, or equivalently (since the limit is a constant), in dist...

One of the most useful generalizations of the central limit theorem is the martingale central limit theorem of Paul Lévy. Lévy was in part inspired by Lindeberg’s treatment of the central limit theorem for sums of independent – but not necessarily identically distributed – random variables. Lindeberg formulated what, in retrospect, is the right hypothesis, now known as the Lindeberg condition,1...

Theorem 1. If μ ∈P(R) has finite kth moment, k ≥ 0, then, writing φ = μ̃: 1. φ ∈ C(R). 2. φ(v) = (i) ∫ R x edμ(x). 3. φ is uniformly continuous. 4. |φ(v)| ≤ ∫ R |x| dμ(x). 1Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, third ed., p. 515, Theorem 15.15; http://individual.utoronto.ca/ jordanbell/notes/narrow.pdf 2Onno van Gaans, Probability measu...