Error Estimates for Certain Probability Limit Theorems

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 $S_n$.