Abstract
The proposed definition of convergence parameter R(W) corresponding to a Markov chain X with a measurable state space (E,ℬ) and any nonempty setW of bounded below measurable functions f: E → ℝ is wider than the well-known definition of convergence parameter R in the sense of Tweedie or Nummelin. Very often, R(W) < ∞, and there exists a set playing the role of the absorbing set inNummelin’s definition ofR. Special attention is paid to the case in whichE is locally compact, X is a Feller chain on E, and W coincides with the family ℰ +0 of all compactly supported continuous functions f ≥ 0 (f ≇ 0). In particular, certain conditions for R(ℰ +0 )−1 to coincide with the norm of an appropriate modification of the chain transition operator are found.
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Original Russian Text © M. G. Shur, 2010, published in Matematicheskie Zametki, 2010, Vol. 87, No. 2, pp. 294–304.
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Shur, M.G. Convergence parameter associated with a Markov chain and a family of functions. Math Notes 87, 271–280 (2010). https://doi.org/10.1134/S0001434610010347
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DOI: https://doi.org/10.1134/S0001434610010347