The moments of first entrance time distributions

Abstract

If f ^* _ij =1, then the sequence {f ⁽ⁿ⁾ _ij , n≧1} determines a discrete probability distribution called the first entrance time distribution from i to j. (For i = j this has also already been called the recurrence time distribution of i in §6.) Thus for each p, $\sum\limits_{N=1}^{\infty }{{{n}^{p}}f\frac{n}{ij}}$ is the moment of order p of this distribution; for p=1 this is the m _ij defined in § 9. More generally, let H be the taboo set; we write

$${{H}^{m\begin{matrix}(p)\\ij\\\end{matrix}}}=\sum\limits_{n=1}^{\infty }{{{n}^{p}}}Hf\begin{matrix}(n)\\ij\\\end{matrix}$$