On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables

Abstract

We present upper bounds of the integral $$ {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathrm{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x $$ for 0 ⩽ l ⩽ 1 + δ, where 0 < δ ⩽ 1, Φ(x) is a standard normal distribution function, and $$ {Z}_N={S}_N/\sqrt{\mathrm{E}{S}_N^2} $$ is the normalized random sum with $$ \mathrm{E}{S}_N^2>0\left({S}_N{X}_1+\dots +{X}_N\right) $$ of centered random variables X1,X2, . . . satisfying the uniformly strong mixing condition. The number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .