### On moderate deviations for martingales

I. G. Grama
Source: Ann. Probab. Volume 25, Number 1 (1997), 152-183.

#### 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.

First Page:
Primary Subjects: 60F10
Secondary Subjects: 60G44
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aop/1024404283
Mathematical Reviews number (MathSciNet): MR1428504
Digital Object Identifier: doi:10.1214/aop/1024404283
Zentralblatt MATH identifier: 0881.60026

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