Laws of Large Numbers for Quadratic Forms, Maxima of Products and Truncated Sums of I.I.D. Random Variables

Abstract

Let $X, X_i$ be i.i.d. real random variables with $EX^2 = \infty$. Necessary and sufficient conditions in terms of the law of $X$ are given for $(1/\gamma_n)\max_{1\leq i<j\leq n} |X_i X_j| \rightarrow 0$ a.s. in general and for $(1/\gamma_n)\sum_{1\leq i\neqj\leq n} X_i X_j \rightarrow 0$ a.s. when the variables $X_i$ are symmetric or regular and the normalizing sequence $\{\gamma_n\}$ is (mildly) regular. The rates of a.s. convergence of sums and maxima of products turn out to be different in general but to coincide under mild regularity conditions on both the law of $X$ and the sequence $\{\gamma_n\}$. Strong laws are also established for $X_{1:n} X_{k:n}$, where $X_{j:n}$ is the $j$th largest in absolute value among $X_1,\ldots,X_n$, and it is found that, under some regularity, the rate is the same for all $k \geq 3$. Sharp asymptotic bounds for $b^{-1}_n \sum^n_{i=1} X_iI_{|X_i|<b_n}$, for $b_n$ relatively small, are also obtained.