On the strong law of large numbers for sums over increasing runs

Given a sequence of n i.i.d. copies of a bivariate random vector (X,Y) with continuously distributed Y, we denote Mn(l),1⩽l⩽n, the maximal sum of the first coordinates over intervals of the length l on which the second coordinates form increasing runs. The strong law of large numbers for the “maximal gain over the longest increasing runs”, i.e. for Mn(Ln), where Ln is the length of the longest increasing run, and n→∞, has been proved in Frolov et al. (Statist. Probab. Lett. 50 (2000) 305) when X has a finite moment of the order r>3. Assuming that X has a finite mean we derive necessary and sufficient conditions for the strong law which turn out to be simple but unusual. We also describe thoroughly the a.s. asymptotics of Ln.