# Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

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## Abstract

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 * E*X

_{1}

^{3+a}I{X

_{1}> 0} < ∞. We derive also some new results for the a.s. asymptotics of L

_{n}. Bibliography: 5 titles.

## Keywords

Asymptotic Behavior Random Vector Indicator Function Maximal Gain Large Index## Preview

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