Abstract
Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the $V$-geometrical ergodicity of a general Markov kernel $P$, which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of $P$ by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in $V$-geometrical ergodicity.
Highlights
Throughout the paper P is a Markov kernel on a measurable space (X, X )
When P admits a unique invariant distribution denoted by π, an important question in the theory of Markov chains is to nd condition for the n−th iterate P n of P to converge to π when n → +∞, and to control kP n − π(·)1X k for some functional norm
Implies that there exists ρ ∈ (0, 1) such that kP n − π(·)1X kV = O(ρn ): this corresponds to the so-called V -geometrical ergodicity property, see [10, 13, 6]
Summary
Throughout the paper P is a Markov kernel on a measurable space (X, X ). Ν(1S ) > 0, we revisit the V -geometrical ergodicity property of P thanks to a simple constructive approach based on an explicit approximation of the iterates of P by positive nite-rank operators, combined with Krein-Rutman theorem [7]. This theorem can be thought of as an abstract dual Perron-Frobenius statement. For the sake of simplicity, let us state the Krein-Rutman theorem for the positive operators on B V In such a context, a proof can be directly obtained from [9, Th 4.1.5, p 251].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
