Operator methods in the problem of estimating the asymptotics of the time of the first hit for the birth and death process (Q1339642)

We study the asymptotic behavior of the time of the first hit of a certain ``high'' level \(n\) for the birth and death process. It is well- known [see \textit{B. V. Gnedenko}, \textit{Yu. K. Belyaev} and \textit{A. D. Solov'ev}, Mathematical methods of reliability theory (1965; Zbl 0146.407)] that the distribution of this time converges to the exponential distribution under fairly general assumptions. We establish the inequalities estimating the rate of this convergence. Our results are mainly obtained by using the operator approach in the ergodic theory of Markov processes. The foundations of this approach are presented in [\textit{N. V. Kartashov}, Theory Probab. Math. Stat. 37, 75-88 (1988), translation from Teor. Veroyatn. Mat. Stat., Kiev 37, 66-77 (1987; Zbl 0659.60102) and the author, ibid. 39, 99-103 (1989), resp. ibid. 39, 83-87 (1988; Zbl 0666.60075)]. Earlier, the problem of esimating the asymptotics of the time of the first hit for the birth and death process was investigated by \textit{A. D. Solov'ev} by using the methods of inverting the Laplace transforms [in: \textit{B. V. Gnedenko} (ed.), Questions of the mathematical theory of reliability (1983; Zbl 0551.60092)]. Similar results for the discrete birth and death process were established by \textit{N. V. Kartashov} [Theory Probab. Appl. 29, 828- 829 (1985); translation from Teor. Veroyatn. Primen. 29, No. 4, 792-793 (1984; Zbl 0607.60053)].