On the random sum with a stopping rule SN > T>

Abstract

We consider the random sum Sn >+ X>1 + X>2 + ...+ Xn >with a stopping rule N>= min{n> : Sn > t}>where T>is a predetermined threshold. We assume that X>1 ,X>2 ,...>are independent and identically distributed random variables having positive integer values. N>is also a positive integer‐valued random variable but dependent on the sequence X>1 , X>2 ,...>due to the stopping rule. Given T>and the probability distribution of X,>we first derive the probability distributions of N, Sn,>and of N>conditional on Sn.>Then we compute the first two moments of these three taking an instructive approach based on the absorbing Markov chain.