LAWS OF LARGE NUMBERS FOR QUADRATIC-FORMS, MAXIMA OF PRODUCTS AND TRUNCATED SUMS OF IID RANDOM-VARIABLES

Let X, X(i) be i.i.d. real random variables with EX(2) = infinity. Necessary and sufficient conditions in terms of the law of X are given for (1/gamma(n)) max(1 less than or equal to i<j less than or equal to n) \X(i)X(j)\ --> 0 a.s. in general and for (1/gamma(n)) Sigma(1 less than or equal to i not equal j less than or equal to n) X(i)X(j) --> 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 jth largest in absolute value among X(1),..., X(n), and it is found that, under some regularity, the rate is the same for all k greater than or equal to 3. Sharp asymptotic bounds for b(n)(-1) Sigma(i=1)(n) X(i)I(\Xi\<bn), for b(n) relatively small, are also obtained.