The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory
Let \((X_ n)\) be an aperiodic Harris recurrent Markov chain on a general state space (S,B), where B is countably generated. Further let m be an initial probability measure, \(P^ n(x,A)\) the transition probabilities of \((X_ n)\) and w the corresponding invariant probability measure. The authors study subgeometrical rates of convergence in Orey's theorem, i.e. the cases where \(\| mP^ n-w\| =o(1/y(n))\) with y belonging to a class of sequences which grow slower than \(r^ n\) (for \(r>1)\). These results are complementary to those obtained by the authors [ibid. 12, 187-202 (1982; Zbl 0484.60056)] and by the first author and \textit{R. L. Tweedie} [Ann. Sci. Univ. Clermont 61, Math. 14(1976), 145-154 (1977; Zbl 0356.60009) and Ann. Probab. 6, 404-420 (1978; Zbl 0378.60051)] for geometric rates of convergence. Results are applied to non-singular renewal measures on \({\mathbb{R}}\). If the conditions \(\int tF(dt)>0\), \(\int y(t)(1-F(t))dt<\infty\) are satisfied for the increment distribution F then a probabilistic method is provided to estimate the right tail of the renewal measure.
geometric - subgeometric rate of convergence
scientific article; zbMATH DE number 3844763