J. Cuzick et al., LAWS OF LARGE NUMBERS FOR QUADRATIC-FORMS, MAXIMA OF PRODUCTS AND TRUNCATED SUMS OF IID RANDOM-VARIABLES, Annals of probability, 23(1), 1995, pp. 292-333
Let X, X(i) be i.i.d. real random variables with EX(2) = infinity. Nec
essary and sufficient conditions in terms of the law of X are given fo
r (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 t
han 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 normaliz
ing sequence {gamma(n)} is (mildly) regular. The rates of a.s. converg
ence of sums and maxima of products turn out to be different in genera
l 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 bound
s for b(n)(-1) Sigma(i=1)(n) X(i)I(\Xi\<bn), for b(n) relatively small
, are also obtained.