• Bernoulli
  • Volume 25, Number 1 (2019), 189-220.

Pólya urns with immigration at random times

Erol Peköz, Adrian Röllin, and Nathan Ross

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We study the number of white balls in a classical Pólya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times are i.i.d. and, under certain moment conditions on the inter-arrival distribution, we characterize the limiting distribution of the (properly scaled) number of white balls as the number of draws goes to infinity. The possible limiting distributions obtained in this way vary considerably depending on the inter-arrival distribution and are difficult to describe explicitly. However, we show that the limits are fixed points of certain probabilistic distributional transformations, and this fact provides a proof of convergence and leads to properties of the limits. The model can alternatively be viewed as a preferential attachment random graph model where added vertices initially have a random number of edges, and from this perspective, our results describe the limit of the degree of a fixed vertex.

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Bernoulli, Volume 25, Number 1 (2019), 189-220.

Received: February 2017
Revised: July 2017
First available in Project Euclid: 12 December 2018

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distributional convergence distributional fixed point equation Pólya urns preferential attachment random graph


Peköz, Erol; Röllin, Adrian; Ross, Nathan. Pólya urns with immigration at random times. Bernoulli 25 (2019), no. 1, 189--220. doi:10.3150/17-BEJ983.

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  • [1] Abramowitz, M. and Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. Washington, DC: For sale by the Superintendent of Documents, U.S. Government Printing Office.
  • [2] Arratia, R., Goldstein, L. and Kochman, F. (2013). Size bias for one and all. Preprint. Available at
  • [3] Barbour, A.D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Science Publications. New York: The Clarendon Press, Oxford Univ. Press.
  • [4] Bartroff, J. and Goldstein, L. (2013). A Berry–Esseen bound for the uniform multinomial occupancy model. Electron. J. Probab. 18 27.
  • [5] Berger, N., Borgs, C., Chayes, J.T. and Saberi, A. (2014). Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Probab. 42 1–40.
  • [6] Brown, M. (2006). Exploiting the waiting time paradox: Applications of the size-biasing transformation. Probab. Engrg. Inform. Sci. 20 195–230.
  • [7] Chauvin, B., Mailler, C. and Pouyanne, N. (2015). Smoothing equations for large Pólya urns. J. Theoret. Probab. 28 923–957.
  • [8] Chauvin, B., Pouyanne, N. and Sahnoun, R. (2011). Limit distributions for large Pólya urns. Ann. Appl. Probab. 21 1–32.
  • [9] Chen, L.H.Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Probability and Its Applications (New York). Heidelberg: Springer.
  • [10] Chen, M.-R. and Kuba, M. (2013). On generalized Pólya urn models. J. Appl. Probab. 50 1169–1186.
  • [11] Chen, M.-R. and Wei, C.-Z. (2005). A new urn model. J. Appl. Probab. 42 964–976.
  • [12] Cooper, C. and Frieze, A. (2003). A general model of web graphs. Random Structures Algorithms 22 311–335.
  • [13] Deijfen, M., van den Esker, H., van der Hofstad, R. and Hooghiemstra, G. (2009). A preferential attachment model with random initial degrees. Ark. Mat. 47 41–72.
  • [14] Dharmadhikari, S.W. and Jogdeo, K. (1969). Bounds on moments of certain random variables. Ann. Math. Stat. 40 1506–1509.
  • [15] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge: Cambridge Univ. Press.
  • [16] Eggenberger, F. and Pólya, G. (1923). Über die statistik verketteter vorgänge. ZAMM Z. Angew. Math. Mech. 3 279–289.
  • [17] Flajolet, P., Gabarró, J. and Pekari, H. (2005). Analytic urns. Ann. Probab. 33 1200–1233.
  • [18] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 177–245.
  • [19] Janson, S. (2006). Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 417–452.
  • [20] Knape, M. and Neininger, R. (2014). Pólya urns via the contraction method. Combin. Probab. Comput. 23 1148–1186.
  • [21] Kuba, M. and Mahmoud, H.M. (2015). Two-color balanced affine urn models with multiple drawings II: Large-index and triangular urns. Preprint. Available at
  • [22] Kuba, M. and Mahmoud, H.M. (2015). Two-colour balanced affine urn models with multiple drawings I: Central limit theorems. Preprint. Available at
  • [23] Laruelle, S. and Pagès, G. (2013). Randomized urn models revisited using stochastic approximation. Ann. Appl. Probab. 23 1409–1436.
  • [24] Mahmoud, H.M. (2009). Pólya Urn Models. Texts in Statistical Science Series. Boca Raton, FL: CRC Press.
  • [25] Pakes, A.G. and Navarro, J. (2007). Distributional characterizations through scaling relations. Aust. N. Z. J. Stat. 49 115–135.
  • [26] Peköz, E., Röllin, A. and Ross, N. (2017). Joint degree distributions of preferential attachment random graphs. Adv. in Appl. Probab. 49 368–387.
  • [27] Peköz, E.A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 1188–1218.
  • [28] Peköz, E.A., Röllin, A. and Ross, N. (2013). Total variation error bounds for geometric approximation. Bernoulli 19 610–632.
  • [29] Peköz, E.A., Röllin, A. and Ross, N. (2016). Generalized gamma approximation with rates for urns, walks and trees. Ann. Probab. 44 1776–1816.
  • [30] Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
  • [31] Petrov, V.V. (1975). Sums of Independent Random Variables. New York–Heidelberg: Springer. Translated from the Russian by A.A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.
  • [32] Pouyanne, N. (2008). An algebraic approach to Pólya processes. Ann. Inst. Henri Poincaré Probab. Stat. 44 293–323.
  • [33] Ross, N. (2013). Power laws in preferential attachment graphs and Stein’s method for the negative binomial distribution. Adv. in Appl. Probab. 45 876–893.
  • [34] Titchmarsh, E.C. (1958). The Theory of Functions. Oxford: Oxford Univ. Press. Reprint of the second (1939) edition.
  • [35] van der Hofstad, R. (2016). Random graphs and complex networks. Final book draft of 1 May, 2016. Available at