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Doeblin trees

François Baccelli 1 Mir-Omid Haji-Mirsadeghi 2 James T. Murphy 3
1 DYOGENE - Dynamics of Geometric Networks
Inria de Paris, CNRS - Centre National de la Recherche Scientifique : UMR 8548, DI-ENS - Département d'informatique de l'École normale supérieure
Abstract : This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a bridge graph, generated by paths started in a fixed state at any time. The bridge graph is made into a unimodular network by marking it and selecting a root in a specified fashion. The unimodularity of this network is leveraged to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e., recurrence both forwards and backwards in time, is introduced and shown to be a key property in uniquely distinguishing paths in the Doeblin graph, and also a decisive property for Markov chains indexed by Z. Properties related to simulating the bridge graph are also studied.
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https://hal.inria.fr/hal-02415283
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Submitted on : Tuesday, December 17, 2019 - 8:37:49 AM
Last modification on : Thursday, June 3, 2021 - 3:07:25 PM

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François Baccelli, Mir-Omid Haji-Mirsadeghi, James T. Murphy. Doeblin trees. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2019, 24, ⟨10.1214/19-EJP375⟩. ⟨hal-02415283⟩

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