A survey of max-type recursive distributional equations

A survey of max-type recursive distributional equations

David J. Aldous and Antar Bandyopadhyay

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

Abstract

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 (ξ_i) and g(⋅) are given and the X_i 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 X_i, are the X_i measurable functions of the innovations process (ξ_i)?

Primary Subjects: 60E05, 62E10, 68Q25, 82B44

Keywords: 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

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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1115137969
Digital Object Identifier: doi:10.1214/105051605000000142
Mathematical Reviews number (MathSciNet): MR2134098
Zentralblatt MATH identifier: 02187024

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