Some limit theorems for large deviations

S. V. Nagaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 2, pp. 231-254

Résumé

Let $\xi_1,\xi_2,\dots$ be a sequence of independent random variables with the same distribution function $F(x)$, $\mathbf E\xi_i=0$ and $\mathbf D\xi_i=1$. Let $F_n(x)$ be the distribution function of $\xi_k$. Let us denote $c_m=\mathbf E|\xi_k|^m$. It is proved that if $c_m\infty$ than the following estimate for $1-F_n(x)$ $$ 1-F_n(x)(1-F(y))+\exp\biggl\{2n\biggl[\frac{m\ln y-\ln nc_mK_m}y\biggr]^2+1\biggr\}\biggl[\frac{nc_mK_m}{y^m}\biggr]^{\frac xy} $$ holds true where $x>0$, $y>0$ and $K_m=\bigl[1+\frac{(m+1)^{m+2}}{e^m}\bigr]$. Then the optimal estimate (in the sense of dependence on $x$) of the remainder term in the central limit theorem when the condition $c_3\infty$ is satisfied is given. Namely we prove that $$ |F_n(x\sqrt n)-\Phi(x)|\frac{Lc_3}{\sqrt n(1+|x|^3)} $$ where $\Phi(x)$ it the standard normal law and $L$ is an absolute constant. Besides Linnik–Petrov's results concerning large deviations are improved. In conclusion an asymptotic expression for the remainder term in the global version of the central limit theorem when $c_3\infty$ is obtained.