Randomly stopped minima and maxima with exponential-type distributions

Let {xi(1), xi(2), . . .} be a sequence of independent real-valued and possibly nonidentically distributed random variables. Suppose that eta is a nonnegative, nondegenerate at 0 and integer-valued random variable, which is independent of {xi(1), xi(2), . . .}. In this paper, we consider conditions for {xi(1), xi(2), . . .} and eta under which the distributions of the randomly stopped maxima and minima, as well as randomly stopped maxima of sums and randomly stopped minima of sums, belong to the class of exponential distributions.