An estimate for an expectation of the simultaneous renewal for time-inhomogeneous Markov chains

Abstract

In this paper, we consider two time-inhomogeneous Markov chains X (l) t , l ∈ {1, 2}, with discrete time on a general state space. We assume the existence of some renewal set C and investigate the time of simultaneous renewal, that is, the first positive time when the chains hit the set C simultaneously. The initial distributions for both chains may be arbitrary. Under the condition of stochastic domination and nonlattice condition for both renewal processes, we derive an upper bound for the expectation of the simultaneous renewal time. Such a bound was calculated for two time-inhomogeneous birth-death Markov chains.

On estimation of expectation of simultaneous renewal time of time-inhomogeneous Markov chains using dominating sequence

“…This estimate involves the second moment of the domination sequence, and in the present paper, we relax that condition and only require the existence of the first moment. We will compare the results of the present paper and [12].…”

“…The critical feature of the proof there was the Daley inequality (see [3]) which gives an estimate for average time of the excess of a renewal process in the homogeneous case. Later, in the paper [13] the Daley inequality was extended to the time-inhomogeneous case which allows obtaining an estimate for simultaneous renewal in the case of a square-integrable dominating sequence for a time-inhomogeneous process (see [12]). This estimate involves the second moment of the domination sequence, and in the present paper, we relax that condition and only require the existence of the first moment.…”

“…Opposing the paper [12] we do not require the existence of the second moment for a dominating sequence.…”

“…We will continue to use the definitions and notations originated in [14] and [12]. Define the renewal intervals…”

“…In the paper [12] we had derived an estimate for a simultaneous renewal of two timeinhomogeneous birth-death processes X (1) and X (2) with the following transition probabilities on the t-th step:…”