• Bernoulli
  • Volume 25, Number 4B (2019), 3623-3651.

Gaps and interleaving of point processes in sampling from a residual allocation model

Jim Pitman and Yuri Yakubovich

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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 Gnedin, Iksanov and Roesler for the Bernoulli sieve.

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Bernoulli, Volume 25, Number 4B (2019), 3623-3651.

Received: April 2018
First available in Project Euclid: 25 September 2019

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GEM distribution interleaving of simple point processes residual allocation model stars and bars duality stationary renewal process Yule process


Pitman, Jim; Yakubovich, Yuri. Gaps and interleaving of point processes in sampling from a residual allocation model. Bernoulli 25 (2019), no. 4B, 3623--3651. doi:10.3150/19-BEJ1104.

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  • [1] Arratia, R., Barbour, A.D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519–535.
  • [2] Athreya, K.B. and Ney, P.E. (1972). Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 176. 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.
  • [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.
  • [5] Borel, É. (1947). Sur les développements unitaires normaux. C. R. Acad. Sci. Paris 225 51.
  • [6] Bruss, F.T. and Rogers, L.C.G. (1991). Pascal processes and their characterization. Stochastic Process. Appl. 37 331–338.
  • [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.
  • [8] Daley, D.J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer Series in Statistics. New York: Springer.
  • [9] Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. in Appl. Probab. 18 1–19.
  • [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.
  • [11] Engen, S. (1975). A note on the geometric series as a species frequency model. Biometrika 62 697–699.
  • [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.
  • [13] Feigin, P.D. (1979). On the characterization of point processes with the order statistic property. J. Appl. Probab. 16 297–304.
  • [14] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed. New York: Wiley.
  • [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.
  • [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.
  • [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.
  • [18] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479.
  • [19] Gnedin, A.V. (2004). The Bernoulli sieve. Bernoulli 10 79–96.
  • [20] Gnedin, A.V., Iksanov, A.M., Negadajlov, P. and Rösler, U. (2009). The Bernoulli sieve revisited. Ann. Appl. Probab. 19 1634–1655.
  • [21] Grandell, J. (1997). Mixed Poisson Processes. Monographs on Statistics and Applied Probability 77. London: CRC Press.
  • [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.
  • [23] Halmos, P.R. (1944). Random alms. Ann. Math. Stat. 15 182–189.
  • [24] Hambly, B. (1992). On the limiting distribution of a supercritical branching process in a random environment. J. Appl. Probab. 29 499–518.
  • [25] Hunt, G.A. (1960). Markoff chains and Martin boundaries. Illinois J. Math. 4 313–340.
  • [26] Ignatov, T. (1982). A constant arising in the asymptotic theory of symmetric groups, and Poisson–Dirichlet measures. Teor. Veroyatn. Primen. 27 129–140.
  • [27] Iksanov, A. (2012). On the number of empty boxes in the Bernoulli sieve II. Stochastic Process. Appl. 122 2701–2729.
  • [28] Iksanov, A. (2013). On the number of empty boxes in the Bernoulli sieve I. Stochastics 85 946–959.
  • [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.
  • [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.
  • [31] Kallenberg, O. (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling 77. Cham: Springer.
  • [32] Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17 373–401.
  • [33] Kendall, D.G. (1966). Branching processes since 1873. J. Lond. Math. Soc. 41 385–406 (1 plate).
  • [34] Lévy, P. (1947). Remarques sur un théorème de M. Émile Borel. C. R. Acad. Sci. Paris 225 918–919.
  • [35] Lundberg, O. (1940). On random processes and their application to sickness and accident statistics. Ph.D. thesis, Univ. Stockholm, Uppsala.
  • [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.
  • [37] Nevzorov, V.B. (2001). Records: Mathematical Theory. Translations of Mathematical Monographs 194. 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 Mathematics 2. Cambridge: Cambridge Univ. Press. Reprint of 1997 original.
  • [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.
  • [40] Pitman, J. and Yor, M. (2001). On the distribution of ranked heights of excursions of a Brownian bridge. Ann. Probab. 29 361–384.
  • [41] Pitman, J.W. (1975). One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 511–526.
  • [42] Rényi, A. (1953). On the theory of order statistics. Acta Math. Acad. Sci. Hung. 4 191–231.
  • [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.
  • [45] Stanley, R.P. (2012). Enumerative Combinatorics, Vol. 1, 2nd ed. Cambridge Studies in Advanced Mathematics 49. Cambridge: Cambridge Univ. Press.
  • [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.
  • [48] Waugh, W.A.O’N. (1970). Transformation of a birth process into a Poisson process. J. Roy. Statist. Soc. Ser. B 32 418–431.