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On the strong law of large numbers for random quadratic forms. (English) Zbl 0844.60011
Theory Probab. Appl. 40, No. 1, 76-91 (1995) and Teor. Veroyatn. Primen. 40, No. 1, 125-142 (1995).
For a given real double sequence \(\{a_{nk}, n \geq 1, k \geq 1\}\), define \(Q_n (x,y) = \sum^n_{i = 1} \sum^n_{j = 1} a_{ij} x_i y_j\), where \(x = \{x_n\}\) and \(y = \{y_n\}\) are any real sequences. Let \(X = \{X_n, n \geq 1\}\) be a sequence of independent random variables and let \(Y = \{Y_n, n \geq 1\}\) be an independent copy of \(X\). This paper presents necessary and sufficient conditions for the quadratic forms \(\{Q_n (X,X)\}\) and for the bilinear forms \(\{Q_n (X, Y)\}\) to obey a strong law of large numbers (i.e., for \(b^{-1}_n Q_n(X,X) \to 0\) a.s. or \(b^{-1}_n Q_n (X,Y) \to 0\) a.s. for some real sequence \(\{b_n\}\)). Special attention is focussed on the cases where the \(X_n\)’s are i.i.d. and symmetric, or are not in the normal domain of partial attraction. In particular, a Prokhorov-type result is established when each \(X_n\) has the standard normal distribution.
60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
60G99 Stochastic processes