Central Limit Theorems for Martingales with Discrete or Continuous Time

Abstract

This survey paper consists of two parts. In the first part (up to and including setion 3) we review the central limit theorems for discrete time martingales, and show that many different sets of conditions for convergence may be reduced to one basic set, where conditions are given in terms of conditional moments of truncated variables, given the past. In the second part (sections 4 and 5) we first recall some basic facts from the modern French theory of stochastic processes, then show that Rebolledo's recent functional limit theorems for martingales with continuous time can be deduced from the limit theorems for discrete time martingales. Again, several related sets of conditions for convergence are formulated.