Probabilities of large deviations for martingales (Q756246)

Consider partial sums \(S_ n=\sum^{n}_{1}X_{ni}\) of a square integrable martingale difference array \((X_{ni},{\mathcal F}_{n0},{\mathcal F}_{ni}:\) \(i=1,...,n)\), \(n=1,2,... \). The author proves that if \[ (i)\quad | X_{ni}| \leq M_ n=o(1)\text{ for all } i=1,...,n;\quad n=1,2,..., \] \[ (ii)\quad \sum^{n}_{1}E(X^ 2_{ni}| {\mathcal F}_{n,i-1})=1\quad a.s.\text{ for all } n=1,2,..., \] then for each function f: (0,\(\infty)\to (0,\infty)\) with f(x)\(\to 0\) as \(x\to \infty\) it holds \[ (1)\quad (1-P(S_ n<x))/(1-\Phi (x))\to 1,\text{ and } (2)\quad P(S_ n<-x)/\Phi (-x)\to 1, \] as \(n\to \infty\) uniformly with respect to \(0\leq x\leq f(n)M_ n^{-1/3}\). Here as usual, \(\Phi\) denotes the standard normal distribution function. Moreover, it is shown that the uniform convergence of (1) and (2) retains for \(0\leq x\leq f(n) \min (M_ n^{-1/3},L_ n^{-1/3})\), if assumption (ii) is replaced by \[ (ii')\quad | \sum^{n}_{1}E(X^ 2_{ni}| {\mathcal F}_{n,i-1})-1| \leq L^ 2_ n=o(1)\quad as\quad n\to \infty. \]