Randomly stopped sums with exponential-type distributions

Abstract

Assume that {ξ1, ξ2, …} are independent and possibly nonidentically distributed random variables. Suppose that η is a nonnegative, nondegenerate at zero and integer-valued random variable, which is independent of {ξ1, ξ2, …}. In this paper, we consider conditions for η and {ξ1, ξ2, …} under which the distribution of the random sum {ξ1 + ξ2 + … + ξη} belongs to the class of exponential distributions.

Keywords

• class of exponential distributions
• random sum
• closure property