Regularly distributed randomly stopped sum, minimum, and maximum

Abstract

Let {ξ₁,ξ₂,...} be a sequence of independent real-valued, possibly nonidentically distributed, random variables, and let η be a nonnegative, nondegenerate at 0, and integer-valued random variable, which is independent of {ξ₁,ξ₂,...}. We consider conditions for {ξ₁,ξ₂,...} and η under which the distributions of the randomly stopped minimum, maximum, and sum are regularly varying.