Convergence to Zero of Quadratic Forms in Independent Random Variables

Abstract

Let $\{X_n\}$ be a sequence of independent random variables none of which are degenerate and define for $y \geqq 0, F(y) = \sup_k P\lbrack |X_k| \geqq y \rbrack$ and $G(y) = \sup_{j \neq k} P\lbrack |X_j X_k| \geqq y \rbrack$. Relationships between the rate of convergence of $F$ and $G$ to zero are investigated. Set $Q_N = \sum_{j, k}a_{jk,N} X_jX_k$ for $N = 1, 2, \cdots$. If the $X$'s are symmetric then it is shown that $Q_N$ converges to zero in probability for a large class of weights $\{a_{jk,N}\}$ if and only if $\lim_{y\rightarrow\infty} yG(y) = 0$. Convergence results are also given for the case when the random variables are not symmetric.