THE ASYMPTOTIC-BEHAVIOR OF LOCALLY SQUARE INTEGRABLE MARTINGALES

Let M be a locally square integrable martingale with predictable quadratic Variance (M) and let Delta M = M - M_ be the jump process of M. In this paper, under the Various restrictions on Delta M, the different increasing rates of M in terms of (M) are obtained. For stochastic integrals X = B . M of the predictable process B with respect to M, the a.s. asymptotic behavior of X is also discussed under restrictions on the rates of increase of B and the restrictions on the conditional distributions of Delta M or on the conditional moments of Delta M. This is applied to some simple examples to determine the convergence rates of estimators in statistics.