A refinement of ergodicity estimates of renewal processes and Markov chains. (English. Russian original) Zbl 0622.60106

Kartashov, N. V.

Theory Probab. Math. Stat. 32, 27-34 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 27-33 (1985).

An estimate is obtained for the generating function of a discrete aperiodic distribution \(q=(q_ t,t\geq 1)\) which separates the values of the function on the unit circle from 1. This estimate leads to refinements of ergodicity inequalities for renewal processes and Markov chains.
For the renewal sequence \(h_ t={\mathbb{P}}(\cup_{n\geq 0}\{\tau_ 1+...+\tau_ n=t\})\), where the \(\tau_ n\) are independent and have distribution q, it is proved, in particular, that for all \(t\geq 1\) \[ t| h_ t-\mu^{-1}-\mu^{-2}\sum_{n>t}Q_ n| \leq \tau^ 2(\omega^ 2+2\omega)/4\mu^ 3, \] where \(\mu =E \tau_ 1\), \(Q_ n={\mathbb{P}}(\tau_ 1>n)\), \(\tau =E \tau_ 1(\tau_ 1-1)\), and \[ \omega \leq 1+\tau \inf (q_ 1^{-1},q_ 2^{-1}+q_ 3^{-1},q_ 5^{- 1}+4q_ 2^{-1},...). \]

MSC:

60K15	Markov renewal processes, semi-Markov processes
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

generating function; ergodicity inequalities; renewal processes