On the distribution tail of the sum of the maxima of two randomly stopped sums in the presence of heavy tails

We deal with two independent random walks with subexponential distributions of their increments. We study the tail distributional asymptotics for the sum of their partial maxima within random time intervals. Assuming the distributions of the lengths of these intervals to be relatively small, with respect to that of the increments of the random walks, we show that the sum of the maxima takes a large value mostly due a large value of a single summand (this is the so-called "principle of a single big jump").