The Asymptotic Behavior of Locally Square Integrable Martingales

Abstract

Let $M$ be a locally square integrable martingale with predictable quadratic variance $\langle M\rangle$ 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 $\langle M\rangle$ are obtained. For stochastic integrals $X = B \cdot M$ 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.