I. G. Grama
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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