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Limit distributions for multitype branching processes of m -ary search trees

Brigitte Chauvin; Quansheng Liu; Nicolas Pouyanne

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 628-654
  • ISSN: 0246-0203

Abstract

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Let m ≥ 3 be an integer. The so-called m -ary search treeis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when m ≤ 26 , the asymptotic behavior of the process is Gaussian, but for m ≥ 27 it is no longer Gaussian and a limit W D T of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of the m -ary search tree. This process satisfies a phase transition of the same kind. In particular, when m ≥ 27 , a limit W of a complex-valued martingale intervenes in its asymptotics. Thanks to the branching property, the law of W satisfies asmoothingequation of the type Z = ℒ e - λ T ( Z ( 1 ) + ⋯ + Z ( m ) ) , where λ is a particular complex number, Z ( k ) are independent complex-valued random variables having the same law as Z , T is a ℝ + -valued random variable independent of the Z ( k ) , and = ℒ denotes equality in law. This distributional equation is extensively studied by various approaches. The existence and uniqueness of solution of the equation are proved by contraction methods. The fact that the distribution of W is absolutely continuous and that its support is the whole complex plane is shown via Fourier analysis. Finally, the existence of exponential moments of W is obtained by considering W as the limit of a complex Mandelbrot cascade.

How to cite

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Chauvin, Brigitte, Liu, Quansheng, and Pouyanne, Nicolas. "Limit distributions for multitype branching processes of $m$-ary search trees." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 628-654. <http://eudml.org/doc/271974>.

@article{Chauvin2014,
abstract = {Let $m\ge 3$ be an integer. The so-called$m$-ary search treeis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m\le 26$, the asymptotic behavior of the process is Gaussian, but for $m\ge 27$ it is no longer Gaussian and a limit $W^\{DT\}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of the $m$-ary search tree. This process satisfies a phase transition of the same kind. In particular, when $m\ge 27$, a limit $W$ of a complex-valued martingale intervenes in its asymptotics. Thanks to the branching property, the law of $W$ satisfies asmoothingequation of the type $Z\stackrel\{ \mathcal \{L\}\}\{=\}\mathrm \{e\}^\{-\lambda T\}(Z^\{(1)\}+\cdots +Z^\{(m)\})$, where $\lambda $ is a particular complex number, $Z^\{(k)\}$ are independent complex-valued random variables having the same law as $Z$, $T$ is a $\mathbb \{R\}_\{+\}$-valued random variable independent of the $Z^\{(k)\}$, and $\stackrel\{ \mathcal \{L\}\}\{=\}$ denotes equality in law. This distributional equation is extensively studied by various approaches. The existence and uniqueness of solution of the equation are proved by contraction methods. The fact that the distribution of $W$ is absolutely continuous and that its support is the whole complex plane is shown via Fourier analysis. Finally, the existence of exponential moments of $W$ is obtained by considering $W$ as the limit of a complex Mandelbrot cascade.},
author = {Chauvin, Brigitte, Liu, Quansheng, Pouyanne, Nicolas},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {martingale; characteristic function; embedding in continuous time; multitype branching process; smoothing transformation; absolute continuity; support; exponential moments; -ary trees; branching processes; Pólya urn; limiting distribution},
language = {eng},
number = {2},
pages = {628-654},
publisher = {Gauthier-Villars},
title = {Limit distributions for multitype branching processes of $m$-ary search trees},
url = {http://eudml.org/doc/271974},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Chauvin, Brigitte
AU - Liu, Quansheng
AU - Pouyanne, Nicolas
TI - Limit distributions for multitype branching processes of $m$-ary search trees
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 628
EP - 654
AB - Let $m\ge 3$ be an integer. The so-called$m$-ary search treeis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m\le 26$, the asymptotic behavior of the process is Gaussian, but for $m\ge 27$ it is no longer Gaussian and a limit $W^{DT}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of the $m$-ary search tree. This process satisfies a phase transition of the same kind. In particular, when $m\ge 27$, a limit $W$ of a complex-valued martingale intervenes in its asymptotics. Thanks to the branching property, the law of $W$ satisfies asmoothingequation of the type $Z\stackrel{ \mathcal {L}}{=}\mathrm {e}^{-\lambda T}(Z^{(1)}+\cdots +Z^{(m)})$, where $\lambda $ is a particular complex number, $Z^{(k)}$ are independent complex-valued random variables having the same law as $Z$, $T$ is a $\mathbb {R}_{+}$-valued random variable independent of the $Z^{(k)}$, and $\stackrel{ \mathcal {L}}{=}$ denotes equality in law. This distributional equation is extensively studied by various approaches. The existence and uniqueness of solution of the equation are proved by contraction methods. The fact that the distribution of $W$ is absolutely continuous and that its support is the whole complex plane is shown via Fourier analysis. Finally, the existence of exponential moments of $W$ is obtained by considering $W$ as the limit of a complex Mandelbrot cascade.
LA - eng
KW - martingale; characteristic function; embedding in continuous time; multitype branching process; smoothing transformation; absolute continuity; support; exponential moments; -ary trees; branching processes; Pólya urn; limiting distribution
UR - http://eudml.org/doc/271974
ER -

References

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