Stochastic Analysis
Proceedings of the Durham Symposium on Stochastic Analysis, 1990
Edited by M. T. Barlow
Edited by N. H. Bingham
Publisher: Cambridge University Press
Print Publication Year: 1991
Online Publication Date:March 2010
Online ISBN:9780511662980
Paperback ISBN:9780521425339
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511662980.003
Subjects: Probability Theory and Stochastic Processes, Abstract Analysis
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INTRODUCTION
Many different models of random trees have arisen in a variety of applied setting, and there is a large but scattered literature on exact and asymptotic results for particular models. For several years I have been interested in what kinds of “general theory” (as opposed to ad hoc analysis of particular models) might be useful in studying asymptotics of random trees. In this paper, aimed at theoretical probabilists, I discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes. No prior knowledge of this subject is assumed: the paper is intended as an introduction and survey.
To give the really big picture in a paragraph, consider a tree on n vertices. View the vertices as points in abstract (rather than d-dimensional) space, but let the edges have length (= 1, as a default) so that there is metric structure: the distance between two vertices is the length of the path between them. Consider the average distance between pairs of vertices. As n → ∞ this average distance could stay bounded or could grow as order n, but almost all natural random trees fall into one of two categories. In the first (and larger) category, the average distance grows as order logn. This category includes supercritical branching processes, and most “Markovian growth” models such as those occurring in the analysis of algorithms. This paper is concerned with the second category, in which the average distance grows as order n½.
pp. i-iv
pp. v-vi
pp. vii-vii
List of participants : Read PDF
pp. viii-viii
An evolution equation for the intersection local times of superprocesses : Read PDF
pp. 1-22
The Continuum random tree II: an overview : Read PDF
pp. 23-70
Harmonic morphisms and the resurrection of Markov processes : Read PDF
pp. 71-90
Statistics of local time and excursions for the Ornstein–Uhlenbeck process : Read PDF
pp. 91-102
LP-Chen forms on loop spaces : Read PDF
pp. 103-162
Convex geometry and nonconfluent Γ-martingales I: tightness and strict convexity : Read PDF
pp. 163-178
Some caricatures of multiple contact diffusion-limited aggregation and the η-model : Read PDF
pp. 179-228
Limits on random measures and stochastic difference equations related to mixing array of random variables : Read PDF
pp. 229-254
Characterizing the weak convergence of stochastic integrals : Read PDF
pp. 255-260
Stochastic differential equations involving positive noise : Read PDF
pp. 261-304
Feeling the shape of a manifold with Brownian motion — the last word in 1990 : Read PDF
pp. 305-320
Decomposition of Dirichlet processes on Hilbert space : Read PDF
pp. 321-332
A supersymmetric Feynman-Kac formula : Read PDF
pp. 333-352
On long excursions of Brownian motion among Poissonian obstacles : Read PDF
pp. 353-375