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April, 1995 The Asymptotic Behavior of Locally Square Integrable Martingales
Jia-Gang Wang
Ann. Probab. 23(2): 552-585 (April, 1995). DOI: 10.1214/aop/1176988279


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.


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Jia-Gang Wang. "The Asymptotic Behavior of Locally Square Integrable Martingales." Ann. Probab. 23 (2) 552 - 585, April, 1995.


Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0831.60053
MathSciNet: MR1334161
Digital Object Identifier: 10.1214/aop/1176988279

Primary: 60F15
Secondary: 60G44 , 60H05 , 62M09

Keywords: Law of the iterated logarithm , locally square integrable martingale , stochastic integral , Strong law of large numbers

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 2 • April, 1995
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