On L 1-weak ergodicity of nonhomogeneous discrete Markov processes and its applications

Abstract

Abstract. In the present paper we investigate the L 1 -weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the L 1 -weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition for such processes to satisfy the L 1 -weak ergodicity. Moreover, we apply the obtained results to establish L 1 -weak ergodicity of discrete time quadratic stochastic processes. As an application of the main result, certain concrete examples are also provided.

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

“…Note that in [15] similar conditions were found for classical nonhomogeneous Markov processes to satisfy weak ergodicity. Applications of such kind of results to quadratic operators can be found in [8,30]. Finally, in Sect.…”

“…In particularly, one can obtain weak ergodicity conditions for classical probability kernels defined on various functional spaces (currently a few papers are devoted to the investigations of Markov chains on metric spaces [34]). Application of such kind of results to quadratic stochastic operators can be found in [8,30].…”

“…Moreover, in this setting mostly, homogeneous Markov processes were considered and investigated (see [4,6,7,16,21,24,35,40]). We should stress that a limited number of papers (see [9,30,47]) were devoted to the investigations of ergodic properties of nonhomogeneous Markov processes in the mentioned scheme. Therefore, this paper is devoted to the study of nonhomogeneous Markov processes in the abstract framework.…”

On stable b-bistochastic quadratic stochastic operators and associated non-homogenous Markov chains

“…Note that, this kind of construction was first considered in [4,20]. Certain properties of the associated Markov chains have been investigated in several paper such as [13,14,19] Let V : S n−1 → S n−1 be a q.s.o. defined by heredity coefficients {P ij,k } n i,j,k=1 and we denote x (m) j = (V (m) (x)) j , x ∈ S n−1 .…”

“…was first considered in [4,20]. Certain properties of the associated Markov chains have been investigated in several paper such as [13,14,19] According to the construction, the Markov measures depend on the initial state of x. Given q.s.o.…”