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January 1997 On moderate deviations for martingales
I. G. Grama
Ann. Probab. 25(1): 152-183 (January 1997). DOI: 10.1214/aop/1024404283

Abstract

Let $X^n = (X_t^n, \mathscr{F}_t^n)_{0 \leq t \leq 1}$ be square integrable martingales with the quadratic characteristics $\langle X^n \rangle, n = 1, 2, \dots$. We prove that the large deviations relation $P(X_1^n \geq r)/(1 - \Phi (r)) \to 1$ holds true for $r$ growing to infinity with some rate depending on $L_{2\delta}^n = E \sum_{0\leq t\leq 1}| \Delta X_t^n |^{2 + 2 \delta}$ and $N_{2 \delta}^n = E | \langle X^n \rangle_1 - 1|^{1 + \delta}$, where $\delta > 0$ and $L_{2 \delta}^n \to 0$, N_{2 \delta}^n \to 0$ as $n \to \infty$. The exact bound for the remainder is also obtained.

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I. G. Grama. "On moderate deviations for martingales." Ann. Probab. 25 (1) 152 - 183, January 1997. https://doi.org/10.1214/aop/1024404283

Information

Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0881.60026
MathSciNet: MR1428504
Digital Object Identifier: 10.1214/aop/1024404283

Subjects:
Primary: 60F10
Secondary: 60G44

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 1 • January 1997
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