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.
Citation
Jack Cuzick. Evarist Gine. Joel Zinn. "Laws of Large Numbers for Quadratic Forms, Maxima of Products and Truncated Sums of I.I.D. Random Variables." Ann. Probab. 23 (1) 292 - 333, January, 1995. https://doi.org/10.1214/aop/1176988388
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