Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

(Homogeneous) Markovian bridges

Vincent Vigon

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Abstract

(Homogeneous) Markov bridges are (time homogeneous) Markov chains which begin at a given point and end at a given point. The price to pay for preserving the homogeneity is to work with processes with a random life-span.

Bridges are studied both for themselves and for their use in describing the transformations of Markov chains: restriction on a random interval, time reversal, time change, various conditionings comprising the confinement in some part of the state space.

These bridges lead us to look at Markov chains from an unusual point of view: we will work, no longer with only one transition matrix, but with a class of matrices which can be deduced one from the other by Doob transformations. This way of proceeding has the advantage of better describing the “past ↔ future symmetries”: The symmetry of conditional independence (well known) and the symmetry of homogeneity (less well known).

Résumé

Les ponts markoviens (homogènes) sont des chaines de Markov (homogènes) qui démarrent à un point donné et meurent à un point donné. Pour préserver l’homogénéité, une telle chaine de Markov a nécessairement une durée de vie aléatoire.

Nous étudions les ponts pour eux mêmes et pour leur utilité à décrire les transformations d’une chaine de Markov : restriction à un intervalle aléatoire, renversement temporel, changement de temps, conditonnements variés : notament le confinement dans une partie de l’espace d’état.

Ces ponts nous conduisent à considérer les chaines de Markov d’un point de vue inhabituel : nous ne travaillons plus avec une seule matrice de transition comme à l’accoutumée, mais avec une classe de matrices qui se déduisent les unes des autres par transformation de Doob. Cette méthode a l’avantage de mieux décrire les symétries passé ↔ futur : symétrie de l’indépendance conditionnelle (bien connue) et symétrie de l’homogénéité (moins bien connue).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 3 (2011), 875-916.

Dates
First available in Project Euclid: 23 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1308834862

Digital Object Identifier
doi:10.1214/10-AIHP391

Mathematical Reviews number (MathSciNet)
MR2841078

Zentralblatt MATH identifier
1267.60080

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 47A68: Factorization theory (including Wiener-Hopf and spectral factorizations) 15A23: Factorization of matrices 60J50: Boundary theory

Keywords
Markov chains Random walks LU-factorization Path-decomposition Fluctuation theory Probabilistic potential theory Infinite matrices Martin boundary

Citation

Vigon, Vincent. (Homogeneous) Markovian bridges. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 3, 875--916. doi:10.1214/10-AIHP391. https://projecteuclid.org/euclid.aihp/1308834862


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