On the Strong Law of Large Numbers for Random Quadratic Forms

Abstract

The paper establishes strong laws of large numbers for the quadratic forms \[ Q_n (X,X) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {a_{ij} X_i X_j } } \] and the bilinear forms \[ Q_n (X,Y) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {a_{ij} X_i Y_j } } , \] where $X = (X_n )$ is a sequence of independent random variables and $Y = (Y_n )$ is an independent copy of it. In the case of independent identically distributed symmetric p-stable random variables $X_n $ we derive necessary and sufficient conditions for the strong laws of $Q_n (X,X)$ and $Q_n (X,Y)$ for a given nondecreasing sequence $(b_n )$ of normalizing constants. For these classes of variables $(X_n )$ the strong laws $\lim b_n^{ - 1} Q_n (X,X) = 0$ a.s. and $\lim b_n^{ - 1} Q_n (X,Y) = 0$ a.s. are shown to be equivalent provided that $a_{ii} = 0$ for all i.