Stochastic fixed-point equations

Abstract The weighted branching process is a natural extension of branching processes. Branching processes count only individuals, weighted branching processes give every individual an abstract weight. Mathematically, we face a (random) dynamical system indexed by a tree. We give an overview of research into this directions, including and unifying many examples of this structure, like Biggins branching random walk and Mandelbrot cascades. WBP are closely connected to stochastic fixed-point equations, fractals and have many applications in genetics, computer science and algorithms. The most advanced example we present is the Quicksort process.

[1]  H. Furstenberg,et al.  Products of Random Matrices , 1960 .

[2]  J. Kahane,et al.  Sur certaines martingales de Benoit Mandelbrot , 1976 .

[3]  J. Biggins,et al.  Martingale convergence in the branching random walk , 1977, Journal of Applied Probability.

[4]  Thomas M. Liggett,et al.  Generalized potlatch and smoothing processes , 1981 .

[5]  R. Durrett,et al.  Fixed points of the smoothing transformation , 1983 .

[6]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[7]  S. Krantz.  Fractal geometry , 1989 .

[8]  Mireille Régnier.  A Limiting Distribution for Quicksort , 1989, RAIRO Theor. Informatics Appl..

[9]  U. Rösler.  A limit theorem for "Quicksort" , 1991, RAIRO Theor. Informatics Appl..

[10]  U. Rösler.  A fixed point theorem for distributions , 1992 .

[11]  Russell Lyons,et al.  Conceptual proofs of L log L criteria for mean behavior of branching processes , 1995 .

[12]  S. Rachev,et al.  Probability metrics and recursive algorithms , 1995, Advances in Applied Probability.

[13]  R. Burton,et al.  An L 2 convergence theorem for random affine mappings , 1995, Journal of Applied Probability.

[14]  Russell Lyons,et al.  A Simple Path to Biggins’ Martingale Convergence for Branching Random Walk , 1998, math/9803100.

[15]  Russell Lyons,et al.  A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes , 1997 .

[16]  Quansheng Liu.  Fixed points of a generalized smoothing transformation and applications to the branching random walk , 1998, Advances in Applied Probability.

[17]  Uwe Rr Osler.  The Weighted Branching Process , 1999 .

[18]  Uwe Rr Osler.  The Contraction Method for Recursive Algorithms , 1999 .

[19]  Uwe Rr Osler.  A Fixed Point Theorem for Distributions , 1999 .

[20]  Quansheng Liu,et al.  On generalized multiplicative cascades , 2000 .

[21]  Svante Janson,et al.  Smoothness and decay properties of the limiting Quicksort density function , 2000, ArXiv.

[22]  Luc Devroye,et al.  Perfect simulation from the Quicksort limit distribution , 2000, ArXiv.

[23]  Svante Janson,et al.  A characterization of the set of fixed points of the Quicksort transformation , 2000, ArXiv.

[24]  A. Caliebe,et al.  Fixed points with finite variance of a smoothing transformation , 2003 .

[25]  Rudolf Grübel,et al.  Asymptotic distribution theory for Hoare's selection algorithm , 1996, Advances in Applied Probability.

[26]  P. Jagers,et al.  Stochastic fixed points for the maximum , 2004 .

[27]  D. Aldous,et al.  A survey of max-type recursive distributional equations , 2004, math/0401388.

[28]  J. D. Biggins,et al.  Measure change in multitype branching , 2004, Advances in Applied Probability.

[29]  L. Rüschendorf,et al.  A general limit theorem for recursive algorithms and combinatorial structures , 2004 .

[30]  L. Rüschendorf,et al.  Analysis of Algorithms by the Contraction Method: Additive and Max-recursive Sequences , 2005 .

[31]  Gerold Alsmeyer,et al.  A stochastic fixed point equation related to weighted branching with deterministic weights. , 2006 .

[32]  R. Neininger,et al.  Probabilistic analysis of algorithms , stochastic fixed-point equations and ideal metrics , 2006 .

[33]  L. Rüschendorf.  On stochastic recursive equations of sum and max type , 2006, Journal of Applied Probability.

[34]  Uwe Rösler,et al.  The contraction method for recursive algorithms , 2001, Algorithmica.

[35]  Gerold Alsmeyer,et al.  A stochastic fixed point equation for weighted minima and maxima , 2008 .

[36]  Mikko Alava,et al.  Branching Processes , 2009, Encyclopedia of Complexity and Systems Science.

[37]  B. Mandelbrot,et al.  Fractional multiplicative processes , 2009, 0902.2902.

[38]  Gerold Alsmeyer,et al.  Fixed points of the smoothing transform: two-sided solutions , 2010, Probability Theory and Related Fields.

[39]  Gerold Alsmeyer,et al.  The functional equation of the smoothing transform , 2009, 0906.3133.

[40]  Henning Sulzbach,et al.  On a functional contraction method , 2012, ArXiv.

[41]  G. Alsmeyer.  The Smoothing Transform: A Review of Contraction Results , 2013 .

[42]  U. Roesler.  Almost sure convergence to the Quicksort process , 2020 .