Compound Geometric Distribution of Order k

The distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T _k be a random variable that follows a geometric distribution of order k, and Y ₁,Y ₂,… a sequence of independent and identically distributed discrete random variables which are independent of T _k . In the present article we develop some results on the distribution of the compound random variable (S_{k} =sum_{t=1}^{T_{k}}Y_{t}).