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Convergence Parameter Associated with a Markov Chain and a Family of Functions
M. G. Shur Moscow State Institute of Electronics and Mathematics (Technical University)
Abstract:
The proposed definition of convergence parameter
$R(W)$ corresponding to a Markov chain
$X$ with a measurable state space
$(E,\mathscr B)$ and any nonempty set
$W$ of bounded below measurable functions
$f\colon E\to\mathbb R$ is wider than the well-known definition of convergence parameter
$R$ in the sense of Tweedie or Nummelin. Very often,
$R(W)<\infty$, and there exists a set playing the role of the absorbing set in Nummelin's definition of
$R$. Special attention is paid to the case in which
$E$ is locally compact,
$X$ is a Feller chain on
$E$, and
$W$ coincides with the family
$\mathscr C_0^+$ of all compactly supported continuous functions
$f\ge 0$ (
$f\not\equiv 0$). In particular, certain conditions for
$R(\mathscr C_0^+)^{-1}$ to coincide with the norm of an appropriate modification of the chain transition operator are found.
Keywords:
convergence parameter, Markov chain, absorbing set, locally compact set, random walk, irreducible chains, Feller chain, measurable state space.
UDC:
519.217.2 Received: 04.04.2008
DOI:
10.4213/mzm4735
Fulltext:
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