Renewal theorem for a class of stationary sequences
Let \(\xi_ n\), \(n\in {\mathbb{Z}}\), be a sequence of stationary r.v. This sequence is assumed to be of the special structure \(\xi_ n=\xi (T^ n(X))\), T the shift operator and \(X=(...,X_{-1},X_ 0,X_ 1,...)\) a sequence of i.i.d. r.v. The main result is an asymptotic renewal theorem for the overshoot \(S_{\tau}-b\) of the partial sums \(S_ n=\sum^{n}_{i=1}\xi_ i\), \(\tau\) the first time n with \(S_ n>b\). The overshoot \(S_{\tau}-b\) and \(X_ 0\) are asymptotically independent as \(b\to \infty.\) This result on stationary sequences includes known results for recurrent Markov chains and processes, functionals of stationary Gaussian processes, and functionals of one-dimensional Gibbs states. The main argument in the proof is a nice coupling argument for the process X.
functionals of stationary Gaussian processes
functionals of one-dimensional Gibbs states
scientific article; zbMATH DE number 3962919
scientific article