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On absorption times and Dirichlet eigenvalues

Laurent Miclo (2010)

ESAIM: Probability and Statistics

This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues. In the discrete time setting, if the underlying Dirichlet eigenvalues (namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on...

On the central limit theorem for some birth and death processes

Tymoteusz Chojecki (2011)

Annales UMCS, Mathematica

Suppose that {Xn, n ≥ 0} is a stationary Markov chain and V is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if [...] [...] converge in law to a normal random variable, as N → +∞. For a stationary Markov chain with the L2 spectral gap the theorem holds for all V such that V (X0) is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables V for...

On the Extinction Probability for Bisexual Branching Processes in Varying Environments

Molina, Manuel, Mota, Manuel, Ramos, Alfonso (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 60J80In this paper, the bisexual branching process in varying environments introduced in [9] is considered and some sufficient conditions for the existence of positive probability of non-extinction are established.Research supported by the Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica, grant BFM2000-0356 and the Consejería de Educación, Ciencia y Tecnología de la Junta de Extremadura and the Fondo Social Europeo, grant IPR00A056....

On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case

Klaus Fleischmann, Vitali Wachtel (2009)

Annales de l'I.H.P. Probabilités et statistiques

Under a well-known scaling, supercritical Galton–Watson processes Z converge to a non-degenerate non-negative random limit variable W. We are dealing with the left tail (i.e. close to the origin) asymptotics of its law. In the Böttcher case (i.e. if always at least two offspring are born), we describe the precise asymptotics exposing oscillations (Theorem 1). Under a reasonable additional assumption, the oscillations disappear (Corollary 2). Also in the Böttcher case, we improve a recent lower deviation...

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