On probabilities of large deviations for sums of independent random variables

L. V. Osipov

Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 2, pp. 320-341

Résumé

Let $X_1,\dots,X_n,\dots$ be a sequence of independent identically distributed random variables with distribution function $F(x)$, and let $\mathbf EX_i=0$, $\mathbf DX_i=1$. Put $$ F_n(x)=\mathbf P\biggl\{\sum_1^nX_i\biggr\},\quad\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-z^2/2}\,dz. $$ Let $\Lambda(z)$ be such a function that $\Lambda(z)/\sqrt z\to\infty$, $z\to\infty$, and $\Lambda(z)$, $1/2\alpha1$. We consider the following problem: under which conditions $$ 1-F_n(x)=\biggl(1-\Phi\Bigl(\frac x{\sqrt n}\Bigr)\biggr)\exp\biggl\{\sum_{\nu=3}^k\mu_\nu\frac{x^\nu}{n^{\nu-1}}\biggr\}(1+o(1)),\quad n\to\infty, $$ uniformly in $x\in[0,\Lambda(n)]$ where $k$ is the largest integer for which $\varlimsup_{z\to\infty}\Lambda^k(z)/z^{k-1}>0$ and $\mu_3,\dots,\mu_k$ are real numbers? Theorem 4 gives an answer to this question under some additional restrictions on $\Lambda(z)$. In Theorem 2 we consider the case $\Lambda(z)=z^\alpha$.