zbMATH — the first resource for mathematics

Characterization and enumeration of certain classes of tenable Pólya urns grown by drawing multisets of balls. (English) Zbl 1339.05007
Summary: We study necessary and sufficient conditions for the strong tenability of Pólya urn schemes under the sampling of multisets of balls. We also investigate sufficient conditions for the tenability (not necessarily in the strong sense) of Pólya urn schemes under the sampling of multisets of balls. We enumerate certain balanced classes and give algorithmic constructions for the replacement matrices for members in the class. We probabilistically analyze the zero-balanced tenable class, and find the asymptotic average proportion of each color, when the starting number of balls is large. We also give an algorithm to determine tenability and construct the Markov chain underlying the scheme, when it is tenable.

05A05 Permutations, words, matrices
05A15 Exact enumeration problems, generating functions
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65C40 Numerical analysis or methods applied to Markov chains
Full Text: DOI
[1] Chen, M; Wei, C, A new urn model, J Appl Probab, 42, 964-976, (2005) · Zbl 1093.60007
[2] Davidson A (2014) Study of the tenability of Pólya urn schemes. Purdue University Dissertation, Purdue, West Lafayette, Indiana · Zbl 0874.68231
[3] Eggenberger, F; Pólya, G, Uber die statistik verketteter vorgänge, Zeitschrift für Angewandte Mathematik und Mechanik, 1, 279-289, (1923) · JFM 49.0382.01
[4] Ehrenfest, P; Ehrenfest, T, Uber zwei bekannte einwände gegen das boltzmannsche H-theorem, Physik, Z, 8, 311-314, (1907) · JFM 38.0931.01
[5] Friedman, B, A simple urn model, Commun Pure Appl Math, 2, 59-70, (1949) · Zbl 0033.07101
[6] Johnson, N; Kotz, S; Mahmoud, H, Pólya-type urn models with multiple drawings, J Iran Stat Soc, 3, 165-173, (2004) · Zbl 06657086
[7] Kholfi, S; Mahmoud, H, The class of tenable zero-balanced Pólya urn schemes: characterization and Gaussian phases, Adv Appl Probab, 44, 702-728, (2012) · Zbl 1269.60008
[8] Knuth D (2009) The art of computer programming, Volume 4A: Combinatorial algorithms, Part 1. Addison-Wesley, Reading, Massachusetts · Zbl 1275.60015
[9] Kotz, S; Balakrishnan, N, Advances in urn models during the past two decades, 203-257, (1997), Boston · Zbl 0888.60014
[10] Kuba M, Mahmoud H (2014) Two-colour balanced affine urn models with multiple drawings I: urns with a small index. Advances in Applied Probability (submitted) · Zbl 0874.68231
[11] Kuba, M; Mahmoud, H; Panholzer, A, Analysis of a generalized friedman’s urn with multiple drawings, Discret Appl Math, 161, 2968-2984, (2013) · Zbl 1292.60012
[12] Mahmoud H (2003) Pólya urn models and connections to random trees: A review. J Iran Stat Soc 53-114 · Zbl 06657072
[13] Mahmoud H (2008) Pólya urn models. Chapman-Hall, Florida · Zbl 1149.60005
[14] Mahmoud, H, Drawing multisets of balls from tenable balanced linear urns, Probab Eng Informational Sci, 27, 147-162, (2013) · Zbl 1275.60015
[15] Mahmoud, H; Smythe, R, Probabilistic analysis of bucket recursive trees, Theor Comput Sci, 144, 221-249, (1995) · Zbl 0874.68231
[16] Mahmoud, H; Tsukiji, T, A limit law for outputs in random recursive circuits, Algorithmica, 31, 403-412, (2000) · Zbl 0989.68107
[17] Mahmoud, H; Smythe, R; Szymański, J, On the structure of plane-oriented recursive trees and their branches, Random Struct Algorithm, 4, 151-176, (1993) · Zbl 0773.05040
[18] Moler, J; Plo, F; Urmeneta, H, A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits, Test, 22, 46-61, (2013) · Zbl 1262.60025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.