## The Annals of Probability

### The Asymptotic Behavior of Locally Square Integrable Martingales

Jia-Gang Wang

#### 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.

#### Article information

Source
Ann. Probab., Volume 23, Number 2 (1995), 552-585.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988279

Digital Object Identifier
doi:10.1214/aop/1176988279

Mathematical Reviews number (MathSciNet)
MR1334161

Zentralblatt MATH identifier
0831.60053

JSTOR

#### Citation

Wang, Jia-Gang. The Asymptotic Behavior of Locally Square Integrable Martingales. Ann. Probab. 23 (1995), no. 2, 552--585. doi:10.1214/aop/1176988279. https://projecteuclid.org/euclid.aop/1176988279