 |
|
|
| | | | | |
|
|
|
|
|
Small counts in the infinite occupancy scheme
|
A. D. Barbour, University of Zurich
A. V. Gnedin, Utrecht University
|
Abstract
The paper is concerned with the classical occupancy
scheme in which balls
are thrown independently into infinitely many
boxes, with given probability of hitting each of the boxes.
We establish joint normal approximation, as the number of balls goes to infinity,
for the numbers of boxes containing any fixed number of balls,
standardized in the natural way, assuming only that the variances of these
counts all tend to infinity. The proof of this approximation is based on a
de-Poissonization lemma. We then review sufficient conditions for the
variances to tend to infinity.
Typically, the normal approximation does not mean convergence.
We show that the convergence of the full vector of counts
only holds under a condition of regular variation, thus giving a complete
characterization of possible limit correlation structures.
|
Full text: PDF
Pages: 365-384
Published on: February 9, 2009
|
Bibliography
- Bentkus, V. A Lyapunov type bound in $Rsp d$. Teor. Veroyatn. Primen. 49 (2004), no. 2, 400--410; translation in Theory Probab. Appl. 49 (2005), no. 2, 311--323 MR2144310 (2006f:60018)
- Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation.Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2 MR0898871 (88i:26004)
- Bogachev, Leonid V.; Gnedin, Alexander V.; Yakubovich, Yuri V. On the variance of the number of occupied boxes.
Adv. in Appl. Math. 40 (2008), no. 4, 401--432. MR2412154
- Le Cam, Lucien. An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10 1960 1181--1197. MR0142174 (25 #5567)
- Chung, Fan; Lu, Linyuan. Concentration inequalities and martingale inequalities: a survey. Internet Math. 3 (2006), no. 1, 79--127. MR2283885 (2008j:60049)
- Gnedin, Alexander; Hansen, Ben; Pitman, Jim. Notes on the occupancy problem with infinitely many boxes: general
asymptotics and power laws.
Probab. Surv. 4 (2007), 146--171 (electronic). MR2318403 (2008g:60056)
- Hwang, Hsien-Kuei; Janson, Svante. Local limit theorems for finite and infinite urn models. Ann. Probab. 36 (2008), no. 3, 992--1022. MR2408581
- Karlin, Samuel. Central limit theorems for certain infinite urn schemes.
J. Math. Mech. 17 1967 373--401. MR0216548 (35 #7379)
- Lamperti, John. An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88 1958 380--387. MR0094863 (20 #1372)
- Louchard, Guy; Prodinger, Helmut; Ward, Mark Daniel. The number of distinct values of some multiplicity in sequences of geometrically distributed random variables. 2005 International Conference on Analysis of Algorithms, 231--256 (electronic), Discrete Math. Theor. Comput. Sci. Proc., AD, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2005. MR2193122
- Michel, R.
Poisson approximation of a nearly homogeneous portfolio.
ASTIN Bulletin 17
1988 165--169.
- Mikhaĭlov, V. G. The central limit theorem for a scheme of independent allocation of particles by cells.(Russian) Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov. 157 (1981), 138--152, 236. MR0651763 (83f:60022)
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context