Asymptotics of the Maximum Sum of Randomly Stopped Finite Random Walks with Subexponential Distribution

We study a finite number of independent random walks with subexponentially distributed increments and negative drifts. We extend the one-dimensional results to finite and fully general stopping times. Assuming that the distribution of the lengths of these intervals is relatively light compared to the distribution of the increments of the random walks, we derive the asymptotic tail distribution of the partial maximum sum over the random time interval.