Characterization of the class of upward first passage time distributions of birth and death processes and related results
Consider a birth and death process \(\{X(t)\}_{t\geq 0}\). Let \(\tau_ n(w)=\inf \{t>0\); \(X(t,w)=n\}\) be the first passage time for X(t) to n and denote by \(\mu_{mn}\) the distribution of \(\tau_ n\) when the process starts at m. Set \({\bar \mu}{}_{mn}=\mu_{mn}/\mu_{mn}([0,\infty))\). Denote by \(ME_+\) the class of mixtures of exponential distributions. Let \(ME_+(0)=ME(0)\) be the class consisting of only one measure - the delta measure at 0. For \(k\geq 1\), denote \(ME_+(k)\) the class of distributions \(\mu\) in \(ME_+\) such that the support of the mixing distribution of \(\mu\) consists of k points in (0,\(\infty)\). For \(k\geq 1\), denote by \(CE_+(k)\) the subclass of probability measure on \(R_+\) consisting of convolutions of k distinct non-degenerate exponential distributions. Let \(\mu_ 1\in CE_+(m)(m\geq 1)\) with Laplace transform \(\prod^{s}_{k=1}a_ k(s+a_ k)^{-1}\) where \(0<a_ 1<a_ 2<...<a_ m<\infty\). For \(0<b_ 1<b_ 2<...<b_ n<\infty\). Let \(\mu_ 2\in ME_+(n)\) (n\(\geq 2)\) with \(\{b_ k\); \(l\leq k\leq n\}\) as the support of its mixing distribution. Then there is a sequence \(\{C_ k\}_{1\leq k\leq n-1}\) such that \(0<b_ 1<C_ 1<b_ 2<...<C_{n- 1}<b_ n\) with the Laplace transform of \(\mu_ 2\) as \[ \prod^{n- 1}_{k=1}C_ k^{-1}(s+C_ k)\prod^{n-1}_{k=1}b_ k(s+b_ k)^{-1}. \] The author calls \(\mu =\mu_ 1*\mu_ 2\) a \(CME_+(m,n)\) distribution if \(\{a_ k\}\cap \{b_ k\}\cap \{c_ k\}=\phi\). He defines classes \(CME_+(m,l)\) for \(m\geq 0\) and \(CME_+(0,n)\) for \(n\geq 2\) by \(CME_+(m,l)=CE_+(m+l)\) and \(CME_+(0,n)=ME_+(n)\). Set \[ CE^ f_+=\cup^{\infty}_{k=l}CE(k)\quad and\quad CME^ f_+=\cup^{\infty}_{m=0}\cup^{\infty}_{n=l}CME_+(m,n). \] The following main theorem and corollary is proved by the author: Theorem 1: Let \(1\leq m\leq n\). Then (i) There is k(max\(\{\) 1,2m-n\(\}\leq k\leq m)\) such that \({\bar \mu}{}_{m,n+l}\in CME_+(n-m,k)\) (ii) For any \(\mu \in CME_+(n-m,k)\) with max\(\{\) 1,2m-n\(\}\leq k\leq m\), there is a birth and death process for which \(\mu_{m,n+l}\) coincides with the class \(CME^ f_+.\) Corollary: The class of first passage time distributions of birth and death processes with reflecting boundary at l coincides with the class \(CME^ f_+\).
scientific article; zbMATH DE number 4090518