Nonlinear renewal theory for Markov random walks
The nonlinear renewal theorem, stating the existence of a limit distribution of the overshoot over level \(a\), is proved for a process \(\{Z_ n, n= 0,1,2,\dots\}\) which is close to a Markov random walk \(\{S_ n\}\) in the sense of closeness of conditional finite-dimensional distributions. This condition follows \textit{M. Woodroofe} [Ann. Probab. 18, No. 4, 1790-1805 (1990; Zbl 0717.60102)] rather than the mainstream of nonlinear renewal theory, e.g. \textit{D. Siegmund} [Sequential analysis. Tests and confidence intervals (1985; Zbl 0573.62071)], where it is assumed that \(Z_ n= S_ n+ \xi_ n\), where the sequence \(\{\xi_ n\}\) satisfies a slow change condition. The Markov random walk is assumed to satisfy the conditions of the Markov renewal theorem of \textit{H. Kesten} [Ann. Probab. 2, 355-386 (1974; Zbl 0303.60090)] involving some continuity assumptions, rather than a Harris chain set-up. An application to a machine breakdown problem is given.
scientific article; zbMATH DE number 713917
scientific article