Laws of large numbers for quadratic forms, maxima of products and truncated sums of i.i.d. random variables
Let \(X_i\) be square-integrable i.i.d. random variables. Necessary and sufficient conditions are given for \({1\over \gamma_n} \max_{1 \leq i < j \leq n} |X_i X_j|\to 0\) a.s. in general and for \({1\over \gamma_n} \sum_{1 \leq i \neq j \leq n} X_i X_j \to 0\) a.s. under some restrictions. Strong laws are obtained for \(X_{1:n} X_{k:n}\), where \(X_{j:n}\) is the \(j\)th largest absolute value among \(X_1, \dots, X_n\). Also, sharp asymptotic bounds are given for \({1\over b_n} \sum^n_{i = 1} X_i \mathbf{ {1}}_{|X_i|< b_n}\) for relatively small \(b_n\).
scientific article; zbMATH DE number 796544