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A probabilistic approach to Carne’s bound. (English) Zbl 1143.60045

Summary: Carne’s bound is a sharp inequality controlling the transition probabilities for a discrete reversible Markov chain (Section 1). Its ordinary proof uses spectral techniques which look as efficient as miraculous. Here we present a new proof, comparing a “drift” for ways “out” and “back”, to get the gaussian part of the bound (Section 2), and using a conditioning technique to get the flight factor (Section 4). Moreover we show how our proof is more “supple” than Carne’s one and may generalize (Section 3.2).

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J35 Transition functions, generators and resolvents
60J45 Probabilistic potential theory

Citations:

Zbl 0584.60078
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References:

[1] Varopoulos, N.Th.: Long range estimates for Markov chains. Bull. Sci. Math. 109(2), 225–252 (1985) · Zbl 0583.60063
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