×

Gaps and interleaving of point processes in sampling from a residual allocation model. (English) Zbl 1428.62190

Summary: This article presents a limit theorem for the gaps \(\widehat{G}_{i:n}:=X_{n-i+1:n}-X_{n-i:n}\) between order statistics \(X_{1:n}\le\cdots\le X_{n:n}\) of a sample of size \(n\) from a random discrete distribution on the positive integers \((P_1,P_2,\ldots)\) governed by a residual allocation model (also called a Bernoulli sieve) \(P_j:=H_j\prod_{i=1}^{j-1}(1-H_i)\) for a sequence of independent random hazard variables \(H_i\) which are identically distributed according to some distribution of \(H\in(0,1)\) such that \(-\log(1-H)\) has a non-lattice distribution with finite mean \(\mu_{\log}\). As \(n\to\infty\) the finite dimensional distributions of the gaps \(\widehat{G}_{i:n}\) converge to those of limiting gaps \(G_i\) which are the numbers of points in a stationary renewal process with i.i.d. spacings \(-\log(1-H_j)\) between times \(T_{i-1}\) and \(T_i\) of births in a Yule process, that is \(T_i:=\sum_{k=1}^i\varepsilon_k/k\) for a sequence of i.i.d. exponential variables \(\varepsilon_k\) with mean 1. A consequence is that the mean of \(\widehat{G}_{i:n}\) converges to the mean of \(G_i\), which is \(1/(i\mu_{\log})\). This limit theorem simplifies and extends a result of A. Gnedin et al. for the Bernoulli sieve [in: Fifth colloquium on mathematics and computer science. Lectures from the colloquium. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). 235–242 (2008; Zbl 1357.60096)].

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
60K05 Renewal theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Citations:

Zbl 1357.60096
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Arratia, R., Barbour, A.D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab.2 519-535. · Zbl 0756.60006 · doi:10.1214/aoap/1177005647
[2] Athreya, K.B. and Ney, P.E. (1972). Branching Processes. Die Grundlehren der Mathematischen Wissenschaften176. New York: Springer.
[3] Birkner, M., Geiger, J. and Kersting, G. (2005). Branching processes in random environment—A view on critical and subcritical cases. In Interacting Stochastic Systems 269-291. Berlin: Springer. · Zbl 1084.60062
[4] Bogachev, L.V., Gnedin, A.V. and Yakubovich, Y.V. (2008). On the variance of the number of occupied boxes. Adv. in Appl. Math.40 401-432. · Zbl 1181.60014 · doi:10.1016/j.aam.2007.05.002
[5] Borel, É. (1947). Sur les développements unitaires normaux. C. R. Acad. Sci. Paris225 51. · Zbl 0029.15303
[6] Bruss, F.T. and Rogers, L.C.G. (1991). Pascal processes and their characterization. Stochastic Process. Appl.37 331-338. · Zbl 0743.60043 · doi:10.1016/0304-4149(91)90052-E
[7] Carmona, P., Petit, F. and Yor, M. (1998). Beta – gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoam.14 311-367. · Zbl 0919.60074 · doi:10.4171/RMI/241
[8] Daley, D.J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer Series in Statistics. New York: Springer. · Zbl 0657.60069
[9] Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. in Appl. Probab.18 1-19. · Zbl 0588.92013 · doi:10.2307/1427237
[10] Duchamps, J.-J., Pitman, J. and Tang, W. Renewal sequences and record chains related to multiple zeta sums. Trans. Amer. Math. Soc. Published online: September 18, 2018. · Zbl 1443.11165 · doi:10.1090/tran/7516
[11] Engen, S. (1975). A note on the geometric series as a species frequency model. Biometrika62 697-699. · Zbl 0331.62076 · doi:10.1093/biomet/62.3.697
[12] Erdős, P., Rényi, A. and Szüsz, P. (1958). On Engel’s and Sylvester’s series. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.1 7-32. · Zbl 0107.27002
[13] Feigin, P.D. (1979). On the characterization of point processes with the order statistic property. J. Appl. Probab.16 297-304. · Zbl 0403.60054 · doi:10.2307/3212898
[14] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed. New York: Wiley. · Zbl 0155.23101
[15] Gnedin, A., Hansen, B. and Pitman, J. (2007). Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws. Probab. Surv.4 146-171. · Zbl 1189.60050 · doi:10.1214/07-PS092
[16] Gnedin, A., Iksanov, A. and Marynych, A. (2010). The Bernoulli sieve: An overview. In 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’10). Discrete Math. Theor. Comput. Sci. Proc., AM 329-341. Nancy: Assoc. Discrete Math. Theor. Comput. Sci. · Zbl 1357.60026
[17] Gnedin, A., Iksanov, A. and Roesler, U. (2008). Small parts in the Bernoulli sieve. In Fifth Colloquium on Mathematics and Computer Science. Discrete Math. Theor. Comput. Sci. Proc., AI 235-242. Nancy: Assoc. Discrete Math. Theor. Comput. Sci. · Zbl 1357.60096
[18] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab.33 445-479. · Zbl 1070.60034 · doi:10.1214/009117904000000801
[19] Gnedin, A.V. (2004). The Bernoulli sieve. Bernoulli10 79-96. · Zbl 1044.60005 · doi:10.3150/bj/1077544604
[20] Gnedin, A.V., Iksanov, A.M., Negadajlov, P. and Rösler, U. (2009). The Bernoulli sieve revisited. Ann. Appl. Probab.19 1634-1655. · Zbl 1178.60019 · doi:10.1214/08-AAP592
[21] Grandell, J. (1997). Mixed Poisson Processes. Monographs on Statistics and Applied Probability77. London: CRC Press. · Zbl 0922.60005
[22] Greenwood, P. and Pitman, J. (1980). Construction of local time and Poisson point processes from nested arrays. J. Lond. Math. Soc. (2) 22 182-192. · Zbl 0427.60048
[23] Halmos, P.R. (1944). Random alms. Ann. Math. Stat.15 182-189. · Zbl 0060.28302 · doi:10.1214/aoms/1177731283
[24] Hambly, B. (1992). On the limiting distribution of a supercritical branching process in a random environment. J. Appl. Probab.29 499-518. · Zbl 0758.60089 · doi:10.2307/3214889
[25] Hunt, G.A. (1960). Markoff chains and Martin boundaries. Illinois J. Math.4 313-340. · Zbl 0094.32103 · doi:10.1215/ijm/1255456049
[26] Ignatov, T. (1982). A constant arising in the asymptotic theory of symmetric groups, and Poisson-Dirichlet measures. Teor. Veroyatn. Primen.27 129-140. · Zbl 0495.60055
[27] Iksanov, A. (2012). On the number of empty boxes in the Bernoulli sieve II. Stochastic Process. Appl.122 2701-2729. · Zbl 1251.60021 · doi:10.1016/j.spa.2012.04.010
[28] Iksanov, A. (2013). On the number of empty boxes in the Bernoulli sieve I. Stochastics85 946-959. · Zbl 1295.60029 · doi:10.1080/17442508.2012.688974
[29] Iksanov, A.M., Marynych, A.V. and Vatutin, V.A. (2015). Weak convergence of finite-dimensional distributions of the number of empty boxes in the Bernoulli sieve. Theory Probab. Appl.59 87-113. · Zbl 1350.60015 · doi:10.1137/S0040585X97986904
[30] Johnson, N.L., Kemp, A.W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd ed. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley Interscience. · Zbl 1092.62010
[31] Kallenberg, O. (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling77. Cham: Springer. · Zbl 1376.60003
[32] Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech.17 373-401. · Zbl 0154.43701
[33] Kendall, D.G. (1966). Branching processes since 1873. J. Lond. Math. Soc.41 385-406 (1 plate). · Zbl 0154.42505 · doi:10.1112/jlms/s1-41.1.385
[34] Lévy, P. (1947). Remarques sur un théorème de M. Émile Borel. C. R. Acad. Sci. Paris225 918-919. · Zbl 0029.15304
[35] Lundberg, O. (1940). On random processes and their application to sickness and accident statistics. Ph.D. thesis, Univ. Stockholm, Uppsala. · Zbl 0063.03678
[36] Neuts, M.F. and Resnick, S.I. (1971). On the times of births in a linear birthprocess. J. Aust. Math. Soc.12 473-475. · Zbl 0232.60076 · doi:10.1017/S1446788700010363
[37] Nevzorov, V.B. (2001). Records: Mathematical Theory. Translations of Mathematical Monographs194. Providence, RI: Amer. Math. Soc. Translated from the Russian manuscript by D.M. Chibisov.
[38] Norris, J.R. (1998). Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics2. Cambridge: Cambridge Univ. Press. Reprint of 1997 original. · Zbl 0938.60058
[39] Pitman, J. and Yakubovich, Y. (2017). Extremes and gaps in sampling from a GEM random discrete distribution. Electron. J. Probab.22 Paper No. 44, 26. · Zbl 1364.60069 · doi:10.1214/17-EJP59
[40] Pitman, J. and Yor, M. (2001). On the distribution of ranked heights of excursions of a Brownian bridge. Ann. Probab.29 361-384. · Zbl 1033.60050 · doi:10.1214/aop/1008956334
[41] Pitman, J.W. (1975). One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab.7 511-526. · Zbl 0332.60055 · doi:10.2307/1426125
[42] Rényi, A. (1953). On the theory of order statistics. Acta Math. Acad. Sci. Hung.4 191-231. · Zbl 0052.14202
[43] Sawyer, S. and Hartl, D. (1985). A sampling theory for local selection. J. Genet.64 21-29.
[44] Smith, W.L. and Wilkinson, W.E. (1969). On branching processes in random environments. Ann. Math. Stat.40 814-827. · Zbl 0184.21103 · doi:10.1214/aoms/1177697589
[45] Stanley, R.P. (2012). Enumerative Combinatorics, Vol. 1, 2nd ed. Cambridge Studies in Advanced Mathematics49. Cambridge: Cambridge Univ. Press. · Zbl 1247.05003
[46] Sukhatme, P.V. (1937). Tests of significance for samples of the \(\chi^2\) -population with two degrees of freedom. Ann. Eugen.8 52-56.
[47] Vervaat, W. (1973). Limit theorems for records from discrete distributions. Stochastic Process. Appl.1 317-334. · Zbl 0273.60014 · doi:10.1016/0304-4149(73)90015-X
[48] Waugh, W.A.O’N. (1970). Transformation of a birth process into a Poisson process. J. Roy. Statist. Soc. Ser. B32 418-431. · Zbl 0219.60067 · doi:10.1111/j.2517-6161.1970.tb00853.x
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.