Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

An algebraic approach to Pólya processes

Nicolas Pouyanne

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Abstract

Pólya processes are natural generalizations of Pólya–Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small when 1 is a simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part ≤1/2; otherwise, it is called large).

Résumé

Les processus de Pólya sont une généralisation naturelle des modèles d’urnes de Pólya–Eggenberger. Cet article présente une nouvelle approche de leur comportement asymptotique via les moments, basée sur la décomposition spectrale d’un opérateur aux différences finies sur des espaces de polynômes. En particulier, elle fournit de nouveaux résultats sur les grands processus (un processus de Pólya est dit petit lorsque 1 est valeur propre simple de sa matrice de remplacement et lorsque toutes les autres valeurs propres ont une partie réelle ≤1/2 ; sinon, on dit qu’il est grand).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 2 (2008), 293-323.

Dates
First available in Project Euclid: 11 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1207948221

Digital Object Identifier
doi:10.1214/07-AIHP130

Mathematical Reviews number (MathSciNet)
MR2446325

Zentralblatt MATH identifier
1185.60029

Subjects
Primary: 60F15: Strong theorems 60F17: Functional limit theorems; invariance principles 60F25: $L^p$-limit theorems 60G05: Foundations of stochastic processes 60G42: Martingales with discrete parameter 60J05: Discrete-time Markov processes on general state spaces 68W40: Analysis of algorithms [See also 68Q25]

Keywords
Pólya processes Pólya–Eggenberger urn processes Strong asymptotics Finite difference transition operator Vector-valued martingale methods

Citation

Pouyanne, Nicolas. An algebraic approach to Pólya processes. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 2, 293--323. doi:10.1214/07-AIHP130. https://projecteuclid.org/euclid.aihp/1207948221


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References

  • K. B. Athreya and S. Karlin. Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. Ann. Math. Statist. 39 (1968) 1801–1817.
  • A. Bagchi and A. K. Pal. Asymptotic normality in the generalized Pólya–Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Methods 6 (1985) 394–405.
  • P. Billingsley. Probability and Measure. Probability and Mathematical Statistics, 2nd edition. Wiley, New York, 1986.
  • B. Chauvin and N. Pouyanne. $m$-ary search trees when $m\geq27$: a strong asymptotics for the space requirements. Random Structures Algorithms 24 (2004) 133–154.
  • H.-H. Chern and H.-K. Hwang. Phase changes in random $m$-ary search trees and generalized quicksort. Random Structures Algorithms 19 (2001) 316–358.
  • H.-H. Chern, M. Fuchs and H.-K. Hwang. Phase changes in random point quadtrees. ACM Trans. Algorithms 3 (2007) Art. 12.
  • F. Eggenberger and G. Pólya. Ueber die Statistik verketter Vorgänge. Z. Angew. Math. Mech. 1 (1923) 279–289.
  • J. A. Fill and N. Kapur. The space requirements of $m$-ary search trees: distributional asymptotics for $m\geq27$. Submitted. Available at.
  • P. Flajolet, J. Gabarró and H. Pekari. Analytic urns. Ann. Probab. 33 (2005) 1200–1233.
  • B. Friedman. A simple urn model. Comm. Pure Appl. Math. 2 (1949) 59–70.
  • R. Gouet. Strong convergence of proportions in a multicolor Pòlya urn. J. Appl. Probab. 34 (1997) 426–435.
  • R. L. Graham, D. E. Knuth and O. Patashnik. Concrete Mathematics, 2nd edition. Addison-Wesley, Reading, MA, 1995.
  • P. Hall and C. C. Heyde. Martingale Limit Theory and Its Applications. Academic Press, New York, 1980.
  • S. Janson. Functional limit theorem for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 (2004) 177–245.
  • S. Janson. Congruence properties of depths in some random trees. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 347–366.
  • S. Janson. Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 (2005) 417–452.
  • G. Pólya. Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré 1 (1930) 117–161.
  • N. Pouyanne. Classification of large Pólya–Eggenberger urns with regard to their asymptotics. In: Internat. Conf. on Analysis of Algorithms. Discrete Math. Theor. Comput. Sci. Proc., AD. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 275–285 (electronic).
  • V. Puyhaubert. Modèles d'urnes et phénomènes de seuils en combinatoire analytique. Thèse de l'Ecole Polytechnique. Available at http://www.imprimerie.polytechnique.fr/Theses/Files/Puyhaubert.pdf, 2005.