On compound sums under dependence

In this paper, we study the compound random variable S=∑t=1NYt when there is a dependence between a random variable N and a sequence of random variables {Yt}t≥1. Such a compound random variable has been found to be useful in several fields including actuarial science, risk management, and reliability. In particular, we develop some results on distributional properties of the random variable S when N is a phase-type random variable that is defined on a sequence of binary trials and depends on {Yt}t≥1. We present illustrative examples and an application for the use of results in actuarial science.