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April, 1965 On the Convergence Rate of the Law of Large Numbers for Linear Combinations of Independent Random Variables
D. L. Hanson, L. H. Koopmans
Ann. Math. Statist. 36(2): 559-564 (April, 1965). DOI: 10.1214/aoms/1177700167

Abstract

The purpose of this paper is to establish the following theorem and several corollaries to it. THEOREM 1. Let $\{\xi_k : k = 0, \pm 1, \cdots\}$ be an independent sequence of real valued random variables with $E\xi_k = 0$ and moment generating functions $f_k(t) = Ee^{t\xi_k}$ such that: (1) for every $\beta > 0$ there exists $T_\beta > 0$ such that $f_k(t)$ exists and $|1 - f_k(t)| \leqq \beta|t|$ for $|t| \leqq T_\beta$ uniformly in $k$. Let $\{a_{n,k} : k = 0, \pm 1, \cdots; n = 1, 2, \cdots\}$ be real numbers such that: (2) $\sum^\infty_{k = -\infty} |a_{n,k}| < A < \infty$ for $n = 1, 2, \cdots$ (3) $f(n) = \sup_k |a_{n,k}| \rightarrow 0$ as $n \rightarrow \infty$. Then $S_n = \lim_{a\rightarrow-\infty,b\rightarrow\infty} \sum^b_{k = a} a_{n,k}\xi_k$ is defined as an almost sure limit for all $n$, and for every $\epsilon > 0$ there exists a positive $\rho_\epsilon < 1$ (depending on $A$ but not on the particular $a_{n,k}'s)$ such that \begin{equation*}\tag{(4)}P\lbrack|S_n| \geqq \epsilon\rbrack \leqq 2\rho_\epsilon^{1/f(n)}.\end{equation*} This theorem is applied in Section 3 to establish exponential convergence rates for the strong law of large numbers for subsequences of linear processes of non-identically distributed random variables. In Section 4, the application of the theorem to the summability theory of sequences of independent random variables is discussed. Section 2 is devoted to proving the theorem.

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D. L. Hanson. L. H. Koopmans. "On the Convergence Rate of the Law of Large Numbers for Linear Combinations of Independent Random Variables." Ann. Math. Statist. 36 (2) 559 - 564, April, 1965. https://doi.org/10.1214/aoms/1177700167

Information

Published: April, 1965
First available in Project Euclid: 27 April 2007

zbMATH: 0132.38604
MathSciNet: MR182043
Digital Object Identifier: 10.1214/aoms/1177700167

Rights: Copyright © 1965 Institute of Mathematical Statistics

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Vol.36 • No. 2 • April, 1965
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