On Convergence Rates in the Central Limit Theorem

Abstract

Let $X_1, X_2, \cdots$ be independent random variables with distribution functions $V_1, V_2, \cdots$, zero means and finite non-zero variances $\sigma_1^2,\sigma_2^2, \cdots$. Set $s_n^2 = \sum^n_1 \sigma_i^2$ and $\Phi(x) = (2\pi)^{-\frac{1}{2}} \int^x_{-\infty}e^{-t{}^2/2} \operatorname{dt}$. Define \begin{equation*}\tag{1.1}\psi_n(c) = \sum^n_1 \int_{|x| > c}x^2 dV _i(x).\end{equation*} According to the well-known Lindeberg-Feller Theorem [1] the condition $s_n^{-2}\psi_n(\xi s_n) \rightarrow 0 \text{as} n \rightarrow \infty\quad\text{for all} \xi > 0$ is both necessary and sufficient in order that $P\lbrack(X_1 + \cdots + X_n)s_n^{-1} \leqq x\rbrack \rightarrow \Phi(x)$ uniformly in $x$ as $n \rightarrow \infty$ and that $\max_{1 \leqq j \leqq n}\sigma_js_n^{-1} \rightarrow 0 \text{as} n \rightarrow \infty.$ Using the method of [3] and [4], it is shown that there exists an absolute constant $K$, independent of $n$ and of the particular sequence $V_1, V_2, \cdots$ such that \begin{equation*}\tag{1.2}\sup_{-\infty<x<\infty}|P\lbrack(X_1 + \cdots + X_n)s_n^{-1} \leqq x\rbrack - \Phi(x)| \leqq Ks_n^{-3} \int^{s_n}_0 \psi_n(u) du.\end{equation*} Some corollaries are deduced and the accuracy of this bound is investigated. Using a truncation scheme, an absolute upper bound is also derived for $\sup_{-\infty<x<\infty}|P\lbrack(X_1 + \cdots + X_n)B_n^{-1} \leqq x\rbrack - \Phi(x)|$, where the assumption of finite variances is now dropped and $B_n$ is a norming constant defined in (4.1).