The Annals of Probability

Generalized gamma approximation with rates for urns, walks and trees

Erol A. Peköz, Adrian Röllin, and Nathan Ross

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study a new class of time inhomogeneous Pólya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein’s method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases of these urn models in recursive constructions of random walk paths and trees, yielding rates of convergence for local time and height statistics of simple random walk paths, as well as for the size of random subtrees of uniformly random binary and plane trees.

Article information

Source
Ann. Probab. Volume 44, Number 3 (2016), 1776-1816.

Dates
Received: September 2013
Revised: February 2015
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1463410032

Digital Object Identifier
doi:10.1214/15-AOP1010

Mathematical Reviews number (MathSciNet)
MR3502594

Zentralblatt MATH identifier
06603562

Subjects
Primary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Secondary: 60E10: Characteristic functions; other transforms 60K99: None of the above, but in this section

Keywords
Generalized gamma distribution Pólya urn model Stein’s method distributional transformations random walk random binary trees random plane trees preferential attachment random graphs

Citation

Peköz, Erol A.; Röllin, Adrian; Ross, Nathan. Generalized gamma approximation with rates for urns, walks and trees. Ann. Probab. 44 (2016), no. 3, 1776--1816. doi:10.1214/15-AOP1010. https://projecteuclid.org/euclid.aop/1463410032.


Export citation

References

  • Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • Arratia, R., Goldstein, L. and Kochman, F. (2013). Size bias for one and all. Preprint. Available at arXiv:1308.2729.
  • 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.
  • Bai, Z. D., Hu, F. and Zhang, L.-X. (2002). Gaussian approximation theorems for urn models and their applications. Ann. Appl. Probab. 12 1149–1173.
  • Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
  • Batir, N. (2008). Inequalities for the gamma function. Arch. Math. (Basel) 91 554–563.
  • Bolthausen, E. (1982). Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 672–688.
  • Borodin, A. N. (1987). On the distribution of random walk local time. Ann. Inst. Henri Poincaré Probab. Stat. 23 63–89.
  • Borodin, A. N. (1989). Brownian local time. Uspekhi Mat. Nauk 44 7–48.
  • Brown, M. (2006). Exploiting the waiting time paradox: Applications of the size-biasing transformation. Probab. Engrg. Inform. Sci. 20 195–230.
  • Chatterjee, S. and Shao, Q.-M. (2011). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21 464–483.
  • Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.
  • Chen, L. H. Y. and Shao, Q.-M. (2005). Stein’s method for normal approximation. In An Introduction to Stein’s Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 1–59. Singapore Univ. Press, Singapore.
  • Chung, K. L. (1976). Excursions in Brownian motion. Ark. Mat. 14 155–177.
  • Chung, K. L. and Hunt, G. A. (1949). On the zeros of $\sum^{n}_{1}\pm1$. Ann. of Math. (2) 50 385–400.
  • Csáki, E. and Mohanty, S. G. (1981). Excursion and meander in random walk. Canad. J. Statist. 9 57–70.
  • Döbler, C. (2012). Stein’s method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the polya urn model. Preprint.
  • Döbler, C. (2012). Stein’s method for the half-normal distribution with applications to limit theorems related to simple random walk. Preprint. Available at arXiv:1303.4592.
  • Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39 1563–1572.
  • Durrett, R. T. and Iglehart, D. L. (1977). Functionals of Brownian meander and Brownian excursion. Ann. Probab. 5 130–135.
  • Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketteter Vorgänge. Z. angew. Math Mech. 3 279–289.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I, 3rd ed. Wiley, New York.
  • Flajolet, P., Gabarró, J. and Pekari, H. (2005). Analytic urns. Ann. Probab. 33 1200–1233.
  • Freedman, D. A. (1965). Bernard Friedman’s urn. Ann. Math. Statist 36 956–970.
  • Friedman, B. (1949). A simple urn model. Comm. Pure Appl. Math. 2 59–70.
  • Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
  • Goldstein, L. and Reinert, G. (2005). Zero biasing in one and higher dimensions, and applications. In Stein’s Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 1–18. Singapore Univ. Press, Singapore.
  • Goldstein, L. and Xia, A. (2006). Zero biasing and a discrete central limit theorem. Ann. Probab. 34 1782–1806.
  • Janson, S. (2006a). Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 417–452.
  • Janson, S. (2006b). Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 139–179.
  • Janson, S. (2006c). Conditioned Galton–Watson trees do not grow. Technical report, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees Combinatorics and Probability.
  • Janson, S. (2012). Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9 103–252.
  • Luk, H. M. (1994). Stein’s method for the Gamma distribution and related statistical applications. Ph.D. thesis, Univ. Southern California.
  • Marchal, P. (2003). Constructing a sequence of random walks strongly converging to Brownian motion. In Discrete Random Walks (Paris, 2003). Discrete Math. Theor. Comput. Sci. Proc., AC 181–190 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
  • Meir, A. and Moon, J. W. (1978). On the altitude of nodes in random trees. Canad. J. Math. 30 997–1015.
  • Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
  • Pakes, A. G. and Khattree, R. (1992). Length-biasing, characterizations of laws and the moment problem. Austral. J. Statist. 34 307–322.
  • Panholzer, A. (2004). The distribution of the size of the ancestor-tree and of the induced spanning subtree for random trees. Random Structures Algorithms 25 179–207.
  • Peköz, E. A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 587–608.
  • Peköz, E. A., Röllin, A. and Ross, N. (2013a). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 1188–1218.
  • Peköz, E. A., Röllin, A. and Ross, N. (2013b). Total variation error bounds for geometric approximation. Bernoulli 19 610–632.
  • Peköz, E., Röllin, A. and Ross, N. (2014). Joint degree distributions of preferential attachment random graphs. Preprint. Available at arXiv:1402.4686.
  • Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
  • Pitman, J. (1999). The distribution of local times of a Brownian bridge. In Séminaire de Probabilités, XXXIII. Lecture Notes in Math. 1709 388–394. Springer, Berlin.
  • Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
  • Pitman, J. and Ross, N. (2012). Archimedes, Gauss, and Stein. Notices Amer. Math. Soc. 59 1416–1421.
  • Raič, M. (2003). Normal approximation with Stein’s method. In Proceedings of the Seventh Young Statisticians Meeting. Metodoloski zvezki, Ljubljana.
  • Reinert, G. (2005). Three general approaches to Stein’s method. In An Introduction to Stein’s Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 183–221. Singapore Univ. Press, Singapore.
  • Rémy, J.-L. (1985). Un procédé itératif de dénombrement d’arbres binaires et son application à leur génération aléatoire. RAIRO Inform. Théor. 19 179–195.
  • Röllin, A. and Ross, N. (2015). Local limit theorems via Landau–Kolmogorov inequalities. Bernoulli 21 851–880.
  • Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210–293.
  • 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.
  • Ross, S. and Peköz, E. (2007). A second course in probability. www.ProbabilityBookstore.com, Boston, MA.
  • Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series 7. IMS, Hayward, CA.
  • Vervaat, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 143–149.
  • Wendel, J. G. (1948). Note on the gamma function. Amer. Math. Monthly 55 563–564.
  • Zhang, L.-X., Hu, F. and Cheung, S. H. (2006). Asymptotic theorems of sequential estimation-adjusted urn models. Ann. Appl. Probab. 16 340–369.
  • Zhang, L.-X., Hu, F., Cheung, S. H. and Chan, W. S. (2011). Immigrated urn models—theoretical properties and applications. Ann. Statist. 39 643–671.