Coincidence properties of birth and death processes

~ (λt + [h)t + o(t) if j = i where Xt > 0 for i > 0, μt > 0 for i > 1 and μ0 > 0. We further assume that Pu(ί) satisfies the foward and backward equation in the usual form. In this paper we restrict attention to the case μ0 — 0 so that when the particle enters the state zero it remains there a random length of time according to an exponential distribution with parameter λ0 and then moves into state one etc. In order to avoid inessential difficulties we assume henceforth that the infinitesimal birth and death rates Xt and μi uniquely determine the process. This is equivalent to the condition Σ~-o (πn + l/λn7Γn) = °° where