On the Poisson Process of Order k Space

2010 Mathematics Subject Classification: 60E05, 62P05.

In this notes, the Poisson process of order k as a compound Poisson process is analyzed. We give a brief review of the distributions of order k. Then, some properties of the Poisson process of order k are given as well as probability mass function and recursion formulas. We then describe the defined process as a compound birth process. As application we consider the standard risk model which counting process is the Poisson process of order k. For the Poisson of order k risk model we derive the joint distribution of the time to ruin and the deficit at ruin. As a limiting case we obtain an equation for the ruin probability. We discuss in detail the particular case of exponentially distributed claims.