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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.
##### MSC:
 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks 60G99 Stochastic processes