On the strong law of large numbers for random quadratic forms

The paper establishes strong laws of large numbers for the quadratic forms $Q_n(X,X)=\sum_{i=1}^n\sum_{j=1}^na_{ij}X_iX_j$ and the bilinear forms $Q_n(X,Y)=\sum_{i=1}^n\sum_{j=1}^na_{ij}X_iY_j$, 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 i.i.d. 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 $i$.