Open Access
2019 Doeblin trees
François Baccelli, Mir-Omid Haji-Mirsadeghi, James T. Murphy III
Electron. J. Probab. 24: 1-36 (2019). DOI: 10.1214/19-EJP375


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 $\mathbb{Z} $. Properties related to simulating the bridge graph are also studied.


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François Baccelli. Mir-Omid Haji-Mirsadeghi. James T. Murphy III. "Doeblin trees." Electron. J. Probab. 24 1 - 36, 2019.


Received: 25 November 2018; Accepted: 16 October 2019; Published: 2019
First available in Project Euclid: 6 November 2019

zbMATH: 07142914
MathSciNet: MR4029423
Digital Object Identifier: 10.1214/19-EJP375

Primary: 05C80
Secondary: 60D05 , 60G10 , 60J10

Keywords: bi-recurrent path , bridge graph , Coupling from the past , Doeblin graph , eternal family tree , Markov chain , unimodular network

Vol.24 • 2019
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