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Invariance Principles for Non-Uniform Random Mappings and Trees

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Asymptotic Combinatorics with Application to Mathematical Physics

Part of the book series: NATO Science Series ((NAII,volume 77))

Abstract

In the context of uniform random mappings of an n-element set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n → ∞ limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study non-uniform cases, in this paper we formulate a sampling invariance principle in terms of iterates of a fixed number of random elements. We show that the sampling invariance principle implies many, but not all, of the distributional limits implied by the functional invariance principle. We give direct verifications of the sampling invariance principle in two successive generalizations of the uniform case, to p-mappings (where elements are mapped to i.i.d. non-uniform elements) and P-mappings (where elements are mapped according to a Markov matrix). We compare with parallel results in the simpler setting of random trees.

Research supported in part by N.S.F. Grants DMS-9970901 and DMS-0071448

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Author information

Authors and Affiliations

  1. Department of Statistics, University of California, 367 Evans Hall # 3860, 94720-3860, Berkeley, CA, USA

    David Aldous & Jim Pitman

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  1. David Aldous
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  2. Jim Pitman
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Editors and Affiliations

  1. I.N.R.I.A., Le Chesnay, France

    Vadim Malyshev

  2. Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

    Anatoly Vershik

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Aldous, D., Pitman, J. (2002). Invariance Principles for Non-Uniform Random Mappings and Trees. In: Malyshev, V., Vershik, A. (eds) Asymptotic Combinatorics with Application to Mathematical Physics. NATO Science Series, vol 77. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0575-3_6

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  • DOI: https://doi.org/10.1007/978-94-010-0575-3_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0793-4

  • Online ISBN: 978-94-010-0575-3

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