The Annals of Applied Probability

The Bernoulli sieve revisited

Alexander V. Gnedin, Alexander M. Iksanov, Pavlo Negadajlov, and Uwe Rösler

Full-text: Open access

Enhanced PDF (213 KB)

Abstract

We consider an occupancy scheme in which “balls” are identified with n points sampled from the standard exponential distribution, while the role of “boxes” is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index Kn* of the last occupied box, the number Kn of occupied boxes, the number Kn, 0 of empty boxes whose index is at most Kn*, the index Wn of the first empty box and the number of balls Zn in the last occupied box. It is shown that the limiting distribution of properly scaled and centered Kn* coincides with that of the number of renewals not exceeding logn. A similar result is shown for Kn and Wn under a side condition that prevents occurrence of very small boxes. The condition also ensures that Kn, 0 converges in distribution. Limiting results for Zn are established under an assumption of regular variation.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 4 (2009), 1634-1655.

Dates
First available in Project Euclid: 27 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1248700630

Digital Object Identifier
doi:10.1214/08-AAP592

Mathematical Reviews number (MathSciNet)
MR2538083

Zentralblatt MATH identifier
1178.60019

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Keywords
Occupancy residual allocation model distributional recursion regenerative composition

Citation

Gnedin, Alexander V.; Iksanov, Alexander M.; Negadajlov, Pavlo; Rösler, Uwe. The Bernoulli sieve revisited. Ann. Appl. Probab. 19 (2009), no. 4, 1634--1655. doi:10.1214/08-AAP592. https://projecteuclid.org/euclid.aoap/1248700630


Export citation

References

  • [1] Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society (EMS), Zürich.
  • [2] Bahadur, R. R. (1971). Some Limit Theorems in Statistics. CBMS Regional Conference Series in Applied Mathematics 4. SIAM, Philadelphia.
    Mathematical Reviews (MathSciNet): MR315820
    Zentralblatt MATH: 0257.62015
  • [3] Barbour, A. D. and Gnedin, A. V. (2006). Regenerative compositions in the case of slow variation. Stochastic Process. Appl. 116 1012–1047.
    Mathematical Reviews (MathSciNet): MR2238612
    Zentralblatt MATH: 1114.60030
    Digital Object Identifier: doi:10.1016/j.spa.2005.12.006
  • [4] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR2253162
  • [5] Bingham, N. H. (1972). Limit theorems for regenerative phenomena, recurrent events and renewal theory. Z. Wahrsch. Verw. Gebiete 21 20–44.
    Mathematical Reviews (MathSciNet): MR353459
    Digital Object Identifier: doi:10.1007/BF00535105
  • [6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1015093
  • [7] Erickson, K. B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 263–291.
    Mathematical Reviews (MathSciNet): MR268976
    Zentralblatt MATH: 0212.51601
    Digital Object Identifier: doi:10.2307/1995628
    JSTOR: links.jstor.org
  • [8] Feller, W. (1949). Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67 98–119.
    Mathematical Reviews (MathSciNet): MR32114
    Zentralblatt MATH: 0039.13301
    Digital Object Identifier: doi:10.2307/1990420
    JSTOR: links.jstor.org
  • [9] Flajolet, P. and Sedgewick, R. (1995). Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theoret. Comput. Sci. 144 101–124.
    Mathematical Reviews (MathSciNet): MR1337755
    Zentralblatt MATH: 0869.68056
    Digital Object Identifier: doi:10.1016/0304-3975(94)00281-M
  • [10] Gnedin, A. V. (2004). The Bernoulli sieve. Bernoulli 10 79–96.
    Mathematical Reviews (MathSciNet): MR2044594
    Digital Object Identifier: doi:10.3150/bj/1077544604
    Project Euclid: euclid.bj/1077544604
  • [11] Gnedin, A. V. (2006). Constrained exchangeable partitions. Discrete Mathematics and Theoretical Computer Science Proc. Ser. A G 391–398.
    Mathematical Reviews (MathSciNet): MR2509650
  • [12] 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.
    Mathematical Reviews (MathSciNet): MR2318403
    Digital Object Identifier: doi:10.1214/07-PS092
    Project Euclid: euclid.ps/1180728778
  • [13] Gnedin, A., Iksanov, A. and Roesler, U. (2008). Small parts in the Bernoulli sieve. Discrete Mathematics and Theoretical Computer Science Proc. Ser. AG 239–246.
    Mathematical Reviews (MathSciNet): MR2508790
  • [14] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479.
    Mathematical Reviews (MathSciNet): MR2122798
    Zentralblatt MATH: 1070.60034
    Digital Object Identifier: doi:10.1214/009117904000000801
    Project Euclid: euclid.aop/1109868588
  • [15] Gnedin, A. and Pitman, J. (2005). Self-similar and Markov composition structures. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 326 59–84.
  • [16] Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for regenerative compositions: Gamma subordinators and the like. Probab. Theory Related Fields 135 576–602.
    Mathematical Reviews (MathSciNet): MR2240701
    Zentralblatt MATH: 1099.60023
    Digital Object Identifier: doi:10.1007/s00440-005-0473-0
  • [17] Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 34 468–492.
    Mathematical Reviews (MathSciNet): MR2223948
    Zentralblatt MATH: 1142.60327
    Digital Object Identifier: doi:10.1214/009117905000000639
    Project Euclid: euclid.aop/1147179979
  • [18] de Haan, L. and Resnick, S. I. (1979). Conjugate Π-variation and process inversion. Ann. Probab. 7 1028–1035.
    Mathematical Reviews (MathSciNet): MR548896
    Zentralblatt MATH: 0428.60033
    Digital Object Identifier: doi:10.1214/aop/1176994895
    Project Euclid: euclid.aop/1176994895
  • [19] Heyde, C. C. (1967). A limit theorem for random walks with drift. J. Appl. Probab. 4 144–150.
    Mathematical Reviews (MathSciNet): MR207061
    Zentralblatt MATH: 0152.16602
    Digital Object Identifier: doi:10.2307/3212307
    JSTOR: links.jstor.org
  • [20] Iksanov, A. and Möhle, M. (2007). A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Comm. Probab. 12 28–35.
    Mathematical Reviews (MathSciNet): MR2407414
    Zentralblatt MATH: 1133.60012
  • [21] Iksanov, A. and Möhle, M. (2008). On the number of jumps of random walks with a barrier. Adv. in Appl. Probab. 40 206–228.
    Mathematical Reviews (MathSciNet): MR2411821
    Zentralblatt MATH: 1157.60041
    Digital Object Identifier: doi:10.1239/aap/1208358893
    Project Euclid: euclid.aap/1208358893
  • [22] Kac, M. (1949). On deviations between theoretical and empirical distributions. Proc. Natl. Acad. Sci. USA 35 252–257.
    Mathematical Reviews (MathSciNet): MR29490
    Zentralblatt MATH: 0033.19303
    Digital Object Identifier: doi:10.1073/pnas.35.5.252
    JSTOR: links.jstor.org
  • [23] Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17 373–401.
    Mathematical Reviews (MathSciNet): MR216548
    Zentralblatt MATH: 0154.43701
  • [24] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR2245368
    Zentralblatt MATH: 1103.60004
  • [25] Whittaker, E. T. and Watson, G. N. (1997). A Course in Modern Analysis. Cambridge Univ. Press, Cambridge.

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more