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.
Publié le : 1995-04-14
Classification:
Strong law of large numbers,
law of the iterated logarithm,
locally square integrable martingale,
stochastic integral,
60F15,
60G44,
60H05,
62M09
@article{1176988279,
author = {Wang, Jia-Gang},
title = {The Asymptotic Behavior of Locally Square Integrable Martingales},
journal = {Ann. Probab.},
volume = {23},
number = {3},
year = {1995},
pages = { 552-585},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988279}
}
Wang, Jia-Gang. The Asymptotic Behavior of Locally Square Integrable Martingales. Ann. Probab., Tome 23 (1995) no. 3, pp. 552-585. http://gdmltest.u-ga.fr/item/1176988279/