Annals of Applied Probability

A survey of max-type recursive distributional equations

David J. Aldous and Antar Bandyopadhyay

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In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form $X\mathop{=}\limits^{d}\,g((\xi_{i},X_{i}),i\geq 1)$. Here (ξi) and g(⋅) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(⋅) is essentially a “maximum” or “minimum” function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process Xi, are the Xi measurable functions of the innovations process (ξi)?

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Ann. Appl. Probab., Volume 15, Number 2 (2005), 1047-1110.

First available in Project Euclid: 3 May 2005

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Primary: 60E05: Distributions: general theory 62E10: Characterization and structure theory 68Q25: Analysis of algorithms and problem complexity [See also 68W40] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Branching process branching random walk cavity method coupling from the past fixed point equation frozen percolation mean-field model of distance metric contraction probabilistic analysis of algorithms probability distribution probability on trees random matching


Aldous, David J.; Bandyopadhyay, Antar. A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 (2005), no. 2, 1047--1110. doi:10.1214/105051605000000142.

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