The Annals of Mathematical Statistics

Error Estimates for Certain Probability Limit Theorems

J. M. Shapiro

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Abstract

Consider a sequence of independent random variables $x_1, x_2, \cdots, x_k, \cdots$ with mean 0 and variance $\sigma^2_k$. Let $S_n = (x_1 + \cdots + x_n)/s_n$ where $s^2_n = \sigma^2_1 + \cdots + \sigma^2_n$. The classical forms of the central limit theorem state that, with certain assumptions, the distribution function $F_n(x)$ approaches the Gaussian distribution $\Phi(x) = \frac{1}{\sqrt{2\pi}} \int^x_{-\infty} e^{-u^{2/2}} du.$ Berry [1] and Essen [3] have studied the behavior of $M_n = \underset{-\infty < x < \infty}\sup |F_n(x) - \Phi(x)|$ and in their main theorems have obtained bounds on $M_n$ which involve the moments of $x_k$ through the third. More generally consider a system of random variables $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ such that for each $n$, the variables $x_{n1}, \cdots, x_{nk_n}$ are independent. Let $S_n = x_{n1} + \cdots + x_{nk_n}$ and again let $F_n(x)$ be the distribution function of $S_n$. From a well known theorem of Khintchine [5] it follows that if the random variables $x_{nk}$ are infinitesimal (i.e., $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P\{|x_{nk}| > \epsilon\} = 0$ for every $\epsilon > 0$) then the class of possible limiting distributions of $F_n(x)$ coincides with the class of infinitely divisible distributions. Let $F(x)$ be any infinitely divisible distribution function and let $M_n = \sup_{-\infty < x < \infty} |F_n(x) - F(x)|$. In this paper we obtain bounds on $M_n$ in the case where $F(x)$ and the $x_{nk}$ have finite second moments. It is shown that under necessary and sufficient conditions for $F_n(x)$ to approach $F(x)$, the bounds on $M_n$ obtained approach zero as $n$ becomes infinite. Throughout the paper, given the system $(x_{nk})$ we shall let $F_{nk}(x), \varphi_{nk}(t), \mu_{nk}$, and $\sigma^2_{nk}$ be the distribution function, characteristic function, mean, and variance respectively of $x_{nk}$, and $F_n(x), \varphi_n(t), \mu_n$, and $\sigma^2_n$ have the same meaning for the random variable $S_n$.

Article information

Source
Ann. Math. Statist., Volume 26, Number 4 (1955), 617-630.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728421

Digital Object Identifier
doi:10.1214/aoms/1177728421

Mathematical Reviews number (MathSciNet)
MR75476

Zentralblatt MATH identifier
0065.11401

JSTOR
links.jstor.org

Citation

Shapiro, J. M. Error Estimates for Certain Probability Limit Theorems. Ann. Math. Statist. 26 (1955), no. 4, 617--630. doi:10.1214/aoms/1177728421. https://projecteuclid.org/euclid.aoms/1177728421


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