Laws of Large Numbers for Quadratic Forms , Maxima of Products

Runninghead: LLN for quadratic forms. Abstract Let X; X i be i.i.d. real random variables with EX 2 =1. Necessary and suucient conditions in terms of the law of X are given for 1 n max 1i<jn jX i X j j!0 a.s. in general and for 1 n P 1i6 =jn X i X j !0 a.s. when the variables X i are symmetric or regular and the normalizing sequence f n g is (mildly) regular. The rates of a.s. convergence of sums and maxima of products turn out to be diierent in general but to coincide under mild regularity conditions on both the law of X and the sequence f n g. 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 ;:::;X n , and it is found that, under some regularity, the rate is the same for all k3. Sharp asymptotic bounds for b ?1 n P n i=1 X i I jX i j<b n , for b n relatively small, are also obtained.