Characterization of the class of upward first passage time distributions of birth and death processes and related results
Makoto Yamazato · Journal of the Mathematical Society of Japan · 1988
We define classes $CME_{+}(m, 1)$ for $m\geqq 0$ and $CME_{+}(0, n)$ for $n\geqq 2$ by $CME_{+}(m, 1)$ $=CE_{+}(m+1)$ and $CME_{+}(0, n)=ME_{+}(n)$ , respectively.SetThe superscript $f$ stands for finite.We will discuss, in a forthcoming paper, extensions of the classes $CE_{+},$ $ME_{+}$ and $CME_{+}$ to distributions on the whole line.So we use the subscript $'+$ in order to denote the classes on the nonnegative real line.The following theorem is our main result.THEOREM 1.Let $1\leqq m\leqq n$ .Then the following hold: (i) There is $k( \max\{1,2m-n\}\leqq k\leqq m)$ such that $\overline{\mu}_{m.n+1}\inCME_{+}(n-m, k)$ .