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1968•The Annals of Mathematical StatisticsOpen access

Almost Sure Convergence of Quadratic Forms in Independent Random Variable

Dale E. Varberg

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Abstract

In [3], we began a study of convergence of quadratic forms in independent random variables. Simultaneously, Fau Dyk Tin and G. E. Silov [2] initiated their study of this problem but restricted to the case of quadratic mean convergence and normal variables. Our aim in this paper is to consider carefully the problem of almost sure convergence (convergence with probability one). Several of our results will generalize well known theorems for series of independent random variables. We shall assume throughout that X1, X2, *** is a sequence of independent real random variables with E(Xk) = 0 and E(Xk2) = 1, k = 1, 2, * .. . Note that we do not assume that the Xk's are identically distributed or place conditions on the higher moments. Let (aJk), j, k = 1, 2, ***, be a real (not necessarily symmetric) matrix and let

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In [3], we began a study of convergence of quadratic forms in independent random variables. Simultaneously, Fau Dyk Tin and G. E. Silov [2] initiated their study of this problem but restricted to the case of quadratic mean convergence and normal variables. Our aim in this paper is to consider carefully the problem of almost sure convergence (convergence with probability one). Several of our results will generalize well known theorems for series of independent random variables. We shall assume throughout that X1, X2, *** is a sequence of independent real random variables with E(Xk) = 0 and E(Xk2) = 1, k = 1, 2, * .. . Note that we do not assume that the Xk's are identically distributed or place conditions on the higher moments. Let (aJk), j, k = 1, 2, ***, be a real (not necessarily symmetric) matrix and let

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Available abstract

In [3], we began a study of convergence of quadratic forms in independent random variables. Simultaneously, Fau Dyk Tin and G. E. Silov [2] initiated their study of this problem but restricted to the case of quadratic mean convergence and normal variables. Our aim in this paper is to consider carefully the problem of almost sure convergence (convergence with probability one). Several of our results will generalize well known theorems for series of independent random variables. We shall assume throughout that X1, X2, *** is a sequence of independent real random variables with E(Xk) = 0 and E(Xk2) = 1, k = 1, 2, * .. . Note that we do not assume that the Xk's are identically distributed or place conditions on the higher moments. Let (aJk), j, k = 1, 2, ***, be a real (not necessarily symmetric) matrix and let

Key concepts: Independent and identically distributed random variables, Mathematics, Random variable, Convergence of random variables, Proofs of convergence of random variables, Convergence (economics), Sequence (biology), Quadratic equation

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