NDLI: Rayleigh processes, real trees, and root growth with re-grafting

Content Provider	Springer Nature Link
Author	Evans, Steven N. Winter, Anita Pitman, Jim
Copyright Year	2005
Abstract	The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N→∞ of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N→∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–Hausdorff distance to metrize the space of rooted compact real trees.Berkeley Statistics Technical Report No. 654 (February 2004), revised October 2004. To appear in Probability Theory and Related Fields.
Ending Page	126
Page Count	46
Starting Page	81
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	1
Volume Number	134
Language	English
Publisher	Springer-Verlag
Publisher Date	2005-03-15
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Real tree Probability measures on topological spaces Continuous-time Markov processes on discrete state spaces Continuum random tree Brownian excursion Mathematical and Computational Physics Probability Theory and Stochastic Processes Branching processes (Galton-Watson, birth-and-death, etc.) Quantitative Finance Hausdorff metric Gromov-Hausdorff metric Statistics for Business/Economics/Mathematical Finance/Insurance Piecewise-deterministic Markov process Operations Research/Decision Theory Mathematical Biology in General Aldous-Broder algorithm
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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