On the central and local limit theorem for martingale difference sequences

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

Central and Local Limit Theories in Markov Dependent Random Variables

On the Central Limit Theorems for Forward and Backward Martingales

Let {Xi}i≥1 be a martingale difference sequence with Xi = Si − Si−1. Under some regularity conditions, we show that (X 1+· · ·+X2 Nn)SNn is asymptotically normal, where {Ni}i≥1 is a sequence of positive integer-valued random variables tending to infinity. In a similar manner, a backward (or reverse) martingale central limit theorem with random indices is provided. Keywords—central limit theorem...

A Central Limit Theorem for Local Martingales with Applications to the Analysis of Longitudinal Data

SUMMARY A functional central limit theorem for a local square integrable martingale with persistent disconti-nuities is given. By persistent discontinuities, it is meant that the martingale has jumps which do not vanish asymptotically. This central limit theorem is motivated by problems in the analysis of longitudinal and life history data.

The Central Limit Theorem for Random Perturbations of Rotations

We prove a functional central limit theorem for stationary random sequences given by the transformations T ;! (x; y) = (2x; y + ! + x) mod 1 on the two-dimensional torus. This result is based on a functional central limit theorem for ergodic stationary martingale diierences with values in a separable Hilbert space of square integrable functions.