On the rate of convergence in the central limit theorem for weakly dependent random variables

A. N. Tihomirov

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 4, pp. 800-818

Résumé

Let $X_1,X_2,\dots$ be a stationary sequence of random variables with $\mathbf EX_1=0$, $\mathbf E|X_1|^3<\infty$. Let \begin{gather*} \sigma^2_n=\mathbf E\biggl(\sum_{j=1}^n X_j\biggr)^2,\qquad F_n(x)=\mathbf P\biggl\{\sigma_n^{-1}\sum_{j=1}^n X_j<x\biggr\}, \\ \Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^x e^{-y^2/2}\,dy,\qquad \Delta_n=\sup|F_n(x)-\Phi(x)|. \end{gather*} We prove that if the sequence $X_n$ satisfies a strong mixing condition and if its mixing coefficient decreases exponentially then $$ \Delta_n=O(n^{-1/2}\ln^2n). $$ For the case of $m$-dependent variables we prove that $$ \Delta_n=O(m^2n^{-1/2}). $$