ASYMPTOTIC BEHAVIOR OF THE MAXIMUM OF SUMS OF I.I.D. RANDOM VARIABLES ALONG MONOTONE BLOCKS

摘要

Let {X_i,Y_i}_(i=1,2,...) be an i.i.d. sequence of bivariate random vectors with P(Y_1 = y) = 0 for all y. Put M_n(j) = max_(0 ≤ k ≤ n-j)(X_(k+1) + ...X_(k+j))I_(k,j), where I_(k,k+j) = I{Y_(k+1) < ··· < Y_(k+j)} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_(k,k+l) = 1 for some k = 0,1,..., n - l. The strong law of large numbers for "the maximal gain over the longest increasing runs," i.e., for M_n(L_n) has been recently derived for the case where X_1 has a finite moment of order 3 + ε, ε > 0. Assuming that X_1 has a finite mean, we prove for any a = 0,1,..., that the s.l.l.n. for M(L_n - a) is equivalent to EX_1~(3+a)I{X_1 > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n.