Bernoulli

  • 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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Abstract

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1569398779

Digital Object Identifier
doi:10.3150/19-BEJ1104

Mathematical Reviews number (MathSciNet)
MR4010967

Zentralblatt MATH identifier
07110150

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

Citation

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. https://projecteuclid.org/euclid.bj/1569398779


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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.
  • [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.