Asymptotic behaviour of maximum of sums of i.i.d. random variables along monotone blocks

Let $\{ X_i,Y_i\}_{i=1,2,\dots }$ be an i.i.d. sequence of bivariate random vectors with $P(Y_1=y)=0$ for all $y$. Put $M_n(j)=\max _{0\le k\le n-j} (X_{k+1}+\dots X_{k+j})I_{k,j},$ where $I_{k,k+j}=I\{Y_{k+1}<\dots<Y_{k+j}\}$ denotes the indicator function for the event in the brackets, $1\le j\le n$. Let $L_n$ be the largest $l\le n$, for which $I_{k,k+l}=1$ for some $k=0,1,\dots,n-l$. 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 of $X_1$ with a finite moment of the order $3+\varepsilon,\varepsilon>0$. Assuming that $X_1$ has a finite mean we prove for any $a=0,1,\dots$, that the s.l.l.n. for $M_{(L_n-a)}$ is equivalent to ${\mathbf E}X_1^{3+a}I\{X_1>0\}<\infty$. We derive also some new results for the a.s. asymptotics of $L_n$.