The Annals of Mathematical Statistics

Convergence Rates for the Law of Large Numbers for Linear Combinations of Exchangeable and $^\ast$-Mixing Stochastic Processes

D. L. Hanson and L. H. Koopmans

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Let $\xi = \{\xi_k \mid k = 0, \pm 1, \cdots\}$ be a sequence of real valued random variables with $E\xi_k \equiv 0$ and let $\{a_{n, k} \mid n = 1, 2, \cdots; k = 0, \pm 1, \cdots\}$ be a doubly indexed sequence of real numbers such that \begin{equation*}\tag{1}\sum_k |a_{n,k}| \leqq 1\quad\text{for} n = 1, 2, \cdots,\end{equation*} and \begin{equation*}\tag{2}\lambda(n) = \max_k |a_{n,k}| \rightarrow 0\quad\text{as} n \rightarrow \infty.\end{equation*} (In (1) we have replaced the upper bound $A < \infty$ of Condition 2 [3] by 1 which clearly entails no loss of generality.) The following result was established in [3]: Theorem A. If $\xi$ is an independent sequence of random variables and if the moment generating functions, $f_k(t)$, of the $\xi_k$'s exist and satisfy the condition; \begin{equation*}\tag{3} for every \beta > 0 there exists T_\beta > 0 such that |t| \leqq T_\beta implies\end{equation*} $f_k(t) \leqq \exp (\beta|t|) uniformly in k,$ then the random variables $S_n = \sum^\infty_{k = -\infty} a_{n,k}\xi_k$ are defined almost surely as limits of the partial sums for each $n$ and for every $\epsilon > 0$ there exists $\rho < 1$ (which depends on $\epsilon$ and $T_\beta$ but not on the particular $a_{n,k}$'s) such that $P\lbrack |S_n| \geqq \epsilon\rbrack \leq 2\rho^{1/\lambda(n)}.$ This result was used to establish exponential convergence rates for the law of large numbers for arbitrary subsequences of linear stochastic processes with absolutely convergent coefficients and for the convergence in probability to zero of Toeplitz means of independent random variables, extending previous results in [1] and [4]. In the present paper we investigate upper bounds on $P\lbrack |S_n| \geqq \epsilon\rbrack$ for two types of discrete parameter stochastic processes, $\xi$, closely related to independent sequences; exchangeable processes and $^\ast$-mixing processes. In Section 2 we establish a basic theorem for exchangeable processes, analogous to Theorem A, which enables us to state conditions leading to upper bounds which tend to zero with $n$ at virtually any sub-exponential rate. In Section 3 these bounds are shown to be sharp in certain important special cases by exhibiting a mixture of normal random variables which actually attains them under the given conditions. In Section 4, the exponential bounds obtained in [3] for independent sequences is shown to carry over to $^\ast$-mixing processes, thus extending a result in [2].

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Ann. Math. Statist., Volume 36, Number 6 (1965), 1840-1852.

First available in Project Euclid: 27 April 2007

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Hanson, D. L.; Koopmans, L. H. Convergence Rates for the Law of Large Numbers for Linear Combinations of Exchangeable and $^\ast$-Mixing Stochastic Processes. Ann. Math. Statist. 36 (1965), no. 6, 1840--1852. doi:10.1214/aoms/1177699814.

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