Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-15T01:30:35.909Z Has data issue: false hasContentIssue false
  • English
  • Français

On martingale tail sums in affine two-color urn models with multiple drawings

Part of: Limit theorems Stochastic processes Combinatorial probability

Published online by Cambridge University Press:  04 April 2017

Markus Kuba  and
Henning Sulzbach
Markus Kuba*
Affiliation:
University of Applied Sciences Technikum Wien
Henning Sulzbach*
Affiliation:
McGill University
*
* Postal address: Institute of Applied Mathematics and Natural Sciences, University of Applied Sciences-Technikum Wien, Höchstädtplatz 5, 1200 Wien, Austria. Email address: [email protected]
** Current address: School of Mathematics, University of Birmingham, BirminghamB15 2TT, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

In two recent works, Kuba and Mahmoud (2015a) and (2015b) introduced the family of two-color affine balanced Pólya urn schemes with multiple drawings. We show that, in large-index urns (urn index between ½ and 1) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new, even in the standard model when only one ball is drawn from the urn in each step (except for the classical Pólya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.

Keywords

Urn modelmartingale central limit theoremlaw of the iterated logarithmlarge-index urntriangular urn

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 96 - 117
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Athreya, K. B. and Karlin, S. (1968).Embedding of urn schemes into continuous time Markov branching processes and related limit theorems.Ann. Math. Statist. 39,1801–1817.Google Scholar
[2] Bagchi, A. and Pal, A. K. (1985).Asymptotic normality in the generalized Pólya–Eggenberger urn model, with an application to computer data structures.SIAM J. Algebraic Discrete Math. 6,394–405.CrossRefGoogle Scholar
[3] Bai, Z. D., Hu, F. and Zhang, L.-X. (2002).Gaussian approximation theorems for urn models and their applications.Ann. Appl. Prob. 12,1149–1173.Google Scholar
[4] Barabási, A.-L. and Albert, R. (1999).Emergence of scaling in random networks.Science 286,509–512.Google Scholar
[5] Chauvin, B., Pouyanne, N. and Sahnoun, R. (2011).Limit distributions for large Pólya urns.Ann. Appl. Prob. 21,1–32.Google Scholar
[6] Chauvin, B., Pouyanne, N. and Mailler, C. (2015).Smoothing equations for large Pólya urns.J. Theoret. Prob. 28,923–957.Google Scholar
[7] Chen, M.-R. and Wei, C.-Z. (2015).A new urn model.J. Appl. Prob. 42,964–976.Google Scholar
[8] Chen, M.-R. and Kuba, M. (2013).On generalized Pólya urn models.J. Appl. Prob. 50,1169–1186.Google Scholar
[9] Chung, F. and Lu, L. (2006).Concentration inequalities and martingale inequalities: a survey.Internet Math. 3,79–127.Google Scholar
[10] Devroye, L. and Janson, S. (2011).Long and short paths in uniform random recursive dags.Ark. Mat. 49,61–77.CrossRefGoogle Scholar
[11] Diaz, J., Serna, M. J., Spirakis, P., Toran, J. and Tsukiji, T. (1994).On the expected depth of Boolean circuits. Tech. Rep. LSI-94-7-R, Universitat Politecnica de Catalunya.Google Scholar
[12] Flajolet, P., Dumas, P. and Puyhaubert, V. (2006).Some exactly solvable models of urn process theory. In Proceedings of Fourth Colloquium on Mathematics and Computer Science, (Discrete Math. Theor. Comput. Sci. Proc. AG),DMTCS,Nancy, pp.59–118.Google Scholar
[13] Flajolet, P., Gabarró, J. and Pekari, H. (2005).Analytic urns.Ann. Prob. 33,1200–1233.CrossRefGoogle Scholar
[14] Freedman, D. A. (1965).Bernard Friedman’s urn.Ann. Math. Statist. 36,965–970.Google Scholar
[15] Fuchs, M. (2015).A note on the quicksort asymptotics.Random Structures Algorithms 46,677–687.Google Scholar
[16] Gouet, R. (1993).Martingale functional central limit theorems for a generalized Pólya urn.Ann. Prob. 21,1624–1639.Google Scholar
[17] Graham, R. L., Knuth, D. E. and Patashnik, O. (1994).Concrete Mathematics.Addison-Wesley,Reading.Google Scholar
[18] Grübel, R. and Kabluchko, Z. (2016).A functional central limit theorem for branching random walks, almost sure weak convergence, and applications to random trees.Ann. Appl. Prob. 26,3659–3698.CrossRefGoogle Scholar
[19] Hall, P. (1978).The convergence of moments in the martingale central limit theorem.Z. Wahrsch. Verw. Gebiete 44,253–260.Google Scholar
[20] Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and Its Application.Academic Press,New York.Google Scholar
[21] Heyde, C. C. (1977).On central limit and iterated logarithm supplements to the martingale convergence theorem.J. Appl. Prob. 14,758–775.Google Scholar
[22] Janson, S. (2004).Functional limit theorems for multitype branching processes and generalized Pólya urns.Stoch. Process. Appl. 110,177–245.Google Scholar
[23] Janson, S. (2006).Limit theorems for triangular urn schemes.Prob. Theory Relat. Fields 134,417–452.Google Scholar
[24] Janson, S. (2010).Moments of Gamma type and the Brownian supremum process area.Prob. Surveys 7,1–52.Google Scholar
[25] Johnson, N. L. and Kotz, S. (1977).Urn Models and Their Application.John Wiley,New York.Google Scholar
[26] Johnson, N. L., Kotz, S. and Mahmoud, H. (2004).Pólya-type urn models with multiple drawings.J. Iran. Statist. Soc. 3,165–173.Google Scholar
[27] Knape, M. and Neininger, R. (2014).Pólya urns via the contraction method.Combin. Prob. Comput. 23,1148–1186.Google Scholar
[28] Konzem, S. and Mahmoud, H. (2016).Characterization and enumeration of certain classes of tenable Pólya urns grown by drawing multisets of balls.Methodol. Comput. Appl. Prob. 18,359–375.Google Scholar
[29] Kuba, M., Mahmoud, H. and Panholzer, A. (2013).Analysis of a generalized Friedman’s urn with multiple drawings.Discrete Appl. Math. 161,2968–2984.Google Scholar
[30] Kuba, M. and Mahmoud, H. (2015a).On urn models with multiple drawings I: urns with a small index. Submitted. Available at https://arxiv.org/abs/1503.09069.Google Scholar
[31] Kuba, M. and Mahmoud, H. (2015b).On urn models with multiple drawings II: large-index and triangular urns. Submitted. Available at https://arxiv.org/abs/1509.09053.Google Scholar
[32] Mahmoud, H. (2008).Pólya Urn Models.Chapman and Hall,Orlando.CrossRefGoogle Scholar
[33] Mahmoud, H. (2013).Drawing multisets of balls from tenable balanced linear urns.Prob. Eng. Inf. Sci. 27,147–162.Google Scholar
[34] Mahmoud, H. (2014).The degree profile in some classes of random graphs that generalize recursive trees.Methodol. Comput. Appl. Prob. 16,527–538.Google Scholar
[35] Moler, J., Plo, F. and Urmeneta, H. (2013).A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits.TEST 22,46–61.Google Scholar
[36] Móri, T. (2005).The maximum degree of the Barabási–Albert random tree.Combin. Prob. Comput. 14,339–348.Google Scholar
[37] Neininger, R. (2015).Refined quicksort asymptotics.Random Structures Algorithms 46,346–361.Google Scholar
[38] Peköz, E. A., Röllin, A. and Ross, N. (2013).Degree asymptotics with rates for preferential attachment random graphs.Ann. Appl. Prob. 23,1188–1218.Google Scholar
[39] Pouyanne, N. (2008).An algebraic approach to Pólya processes.Ann. Inst. H. Poincaré Prob. Statist. 44,293–323.Google Scholar
[40] Renlund, H. (2010).Generalized Pólya urns via stochastic approximation. Preprint. Available at https://arxiv.org/abs/1002.3716.Google Scholar
[41] Rónyi, A. and Róvósz, P. (1958).On mixing sequences of random variables.Acta Math. Acad. Sci. H. 9,389–393.Google Scholar
[42] Sulzbach, H. (2016).On martingale tail sums for the path length in random trees. To appear in Random Structures Algorithms. Available at http://dx.doi.org/10.1002/rsa.20674.Google Scholar
[43] Tsukiji, T. and Xhafa, F. (1996).On the depth of randomly generated circuits. In Algorithms—ESA ’96 (Lecture Notes Comput. Sci 1136).Springer,Berlin, pp.208–220.Google Scholar
[44] Tsukiji, T. and Mahmoud, H. (2001).A limit law for outputs in random circuits.Algorithmica 31,403–412.CrossRefGoogle Scholar