The Annals of Mathematical Statistics

On Convergence Rates in the Central Limit Theorem

Ellen S. Hertz

Full-text: Open access


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).

Article information

Ann. Math. Statist., Volume 40, Number 2 (1969), 475-479.

First available in Project Euclid: 27 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Hertz, Ellen S. On Convergence Rates in the Central Limit Theorem. Ann. Math. Statist. 40 (1969), no. 2, 475--479. doi:10.1214/aoms/1177697713.

Export citation