On the Convergence Rate of the Law of Large Numbers for Linear Combinations of Independent Random Variables

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.