Asymptotic behavior of a renewal process thinned by an alternating process (Q1899537)

The exponential asymptotic behavior of the probabilities of rare events in regeneration processes has been considered by \textit{A. D. Solov'ev} [in: Topics of mathematical reliability theory, 9-112 (Moscow, 1983)], \textit{I. N. Kovalenko} [``Analysis of rare events in estimating the efficiency and reliability of systems'' (1980; Zbl 0503.60093)], and \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)]. The method of phase aggregation of semi-Markov systems has been applied by \textit{V. S. Korolyuk} and \textit{A. F. Turbin} [``Markov renewal processes in problems of systems reliability'' (1982; Zbl 0508.60073)]to solve similar problems for some processes of other classes, including the superposition of an alternating process with a renewal process. It has been shown, in particular, that under certain conditions on the initial distribution functions of interrenewal times the limit distribution (after appropriate normalization) of the first arrival of the renewal process in a certain interval of the alternating process is a power-function distribution. In this paper, we solve the same problem in a slightly different setting under different, sufficiently general conditions on the initial distribution functions. We use a new method of analysis of the ``incomplete renewal'' equation, which also produces bounds on the rate of convergence.