Abstract
The definition of the Bernoulli sieve which is an infinite allocation scheme can be found on p. 1. Assuming that the number of balls to be allocated equals n (in other words, using a sample of size n from a uniform distribution on [0, 1]), denote by K n the number of occupied boxes and by M n the index of the last occupied box. Also, put L n : = M n − K n and note that L n equals the number of empty boxes within the occupancy range (i.e., we only count the empty boxes with indices not exceeding M n ).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
G. Alsmeyer, A. Iksanov and A. Marynych, Functional limit theorems for the number of occupied boxes in the Bernoulli sieve. Stoch. Proc. Appl., to appear (2017).
R. R. Bahadur, On the number of distinct values in a large sample from an infinite discrete distribution. Proc. Nat. Inst. Sci. India. 26A (1960), 66–75.
A. D. Barbour, Univariate approximations in the infinite occupancy scheme. Alea, Lat. Am. J. Probab. Math. Stat. 6 (2009), 415–433.
A. D. Barbour and A. V. Gnedin, Small counts in the infinite occupancy scheme. Electron. J. Probab. 14 (2009), 365–384.
L. V. Bogachev, A. V. Gnedin and Yu. V. Yakubovich, On the variance of the number of occupied boxes. Adv. Appl. Math. 40 (2008), 401–432.
D. A. Darling, Some limit theorems assiciated with multinomial trials. Proc. Fifth Berkeley Symp. on Math. Statist. and Probab. 2 (1967), 345–350.
O. Durieu and Y. Wang, From infinite urn schemes to decompositions of self-similar Gaussian processes. Electron. J. Probab. 21 (2016), paper no. 43, 23 pp.
M. Dutko, Central limit theorems for infinite urn models. Ann. Probab. 17 (1989), 1255–1263.
Sh. K. Formanov and A. Asimov, A limit theorem for the separable statistic in a random assignment scheme. J. Sov. Math. 38 (1987), 2405–2411.
A. V. Gnedin, The Bernoulli sieve. Bernoulli 10 (2004), 79–96.
A. Gnedin, A. Hansen and J. Pitman, Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws. Probab. Surv. 4 (2007), 146–171.
A. Gnedin and A. Iksanov, Regenerative compositions in the case of slow variation: A renewal theory approach. Electron. J. Probab. 17 (2012), paper no. 77, 19 pp.
A. Gnedin, A. Iksanov and A. Marynych, Limit theorems for the number of occupied boxes in the Bernoulli sieve. Theory Stochastic Process. 16(32) (2010), 44–57.
A. Gnedin, A. Iksanov, and A. Marynych, The Bernoulli sieve: an overview. In Proceedings of the 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’10), Discrete Math. Theor. Comput. Sci. AM (2010), 329–341.
A. Gnedin, A. Iksanov and A. Marynych, A generalization of the Erdős-Turán law for the order of random permutation. Combin. Probab. Comput. 21 (2012), 715–733.
A. Gnedin, A. Iksanov, P. Negadailov and U. Rösler, The Bernoulli sieve revisited. Ann. Appl. Probab. 19 (2009), 1634–1655.
A. Gnedin, A. Iksanov and U. Roesler, Small parts in the Bernoulli sieve. In Proceedings of the Fifth Colloquium on Mathematics and Computer Science, Discrete Math. Theor. Comput. Sci. Proc. AI (2008), 235–242.
H. K. Hwang and S. Janson, Local limit theorems for finite and infinite urn models. Ann. Probab. 36 (2008), 992–1022.
A. Iksanov, On the number of empty boxes in the Bernoulli sieve II. Stoch. Proc. Appl. 122 (2012), 2701–2729.
A. Iksanov, On the number of empty boxes in the Bernoulli sieve I. Stochastics. 85 (2013), 946–959.
A. M. Iksanov, A. V. Marynych and V. A. Vatutin, Weak convergence of finite-dimensional distributions of the number of empty boxes in the Bernoulli sieve. Theory Probab. Appl. 59 (2015), 87–113.
S. Karlin, Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17 (1967), 373–401.
V. F. Kolchin, B. A. Sevastyanov and V. P. Chistyakov, Random allocations. V.H.Winston & Sons, 1978.
V. G. Mikhailov, The central limit theorem for a scheme of independent allocation of particles by cells. Proc. Steklov Inst. Math. 157 (1983), 147–163.
Sh. A. Mirakhmedov, Randomized decomposable statistics in a generalized allocation scheme over a countable set of cells. Diskret. Mat. 1 (1989), 46–62 (in Russian).
Sh. A. Mirakhmedov, Randomized decomposable statistics in a scheme of independent allocation of particles into cells. Diskret. Mat. 2 (1990), 97–111 (in Russian).
P. Negadailov, Limit theorems for random recurrences and renewal-type processes. PhD thesis, University of Utrecht, the Netherlands. Available at http://igitur-archive.library.uu.nl/dissertations/2010-0823-200228/negadailov.pdf
E. T. Whittaker and G. N. Watson, A course of modern analysis. 4th Edition reprinted, Cambridge University Press, 1950.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this chapter
Cite this chapter
Iksanov, A. (2016). Application to the Bernoulli Sieve. In: Renewal Theory for Perturbed Random Walks and Similar Processes. Probability and Its Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49113-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-49113-4_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-49111-0
Online ISBN: 978-3-319-49113-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)