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