Higher order moments of renewal counting processes and Eulerian polynomials

In this paper we provide explicit formulas for the moments of all orders of Nm, the number of renewals in [0,m] of discrete-time renewal processes. We decompose the sample space of the process, and the formulas are obtained using combinatorial methods. We obtain explicit formulas for the kth order moments in this manner, and we also provide asymptotic results in both the transient and recurrent cases.