The Annals of Probability

The Asymptotic Behavior of Locally Square Integrable Martingales

Jia-Gang Wang

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F15: Strong theorems
Secondary: 60G44: Martingales with continuous parameter 60H05: Stochastic integrals 62M09: Non-Markovian processes: estimation

Strong law of large numbers law of the iterated logarithm locally square integrable martingale stochastic integral


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

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