ON THE CENTRAL AND LOCAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE SEQUENCES

On the central and local limit theorem for martingale difference sequences

Let (Ω,A, μ) be a Lebesgue space and T : Ω → Ω an ergodic measure preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A simila...

The Martingale Central Limit Theorem

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...

A Martingale Central Limit Theorem

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

Martingales, Lévy’s continuity theorem, and the martingale central limit theorem

Exactness of Martingale Approximation and the Central Limit Theorem

Let (X i) be a Markov chain with kernel Q, f an L 2 function on its state space.