Higher order moments of renewal counting processes and Eulerian polynomials

In this paper we provide explicit formulas for the moments of all orders of NmNm, the number of renewals in [0,m][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 kkth order moments E[Nmk] in this manner, and we also provide asymptotic results in both the transient and recurrent cases.