The Annals of Applied Probability

The Bernoulli sieve revisited

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

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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

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

First available in Project Euclid: 27 July 2009

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Zentralblatt MATH identifier

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

Occupancy residual allocation model distributional recursion regenerative composition


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.

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  • [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.
  • [3] Barbour, A. D. and Gnedin, A. V. (2006). Regenerative compositions in the case of slow variation. Stochastic Process. Appl. 116 1012–1047.
  • [4] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [5] Bingham, N. H. (1972). Limit theorems for regenerative phenomena, recurrent events and renewal theory. Z. Wahrsch. Verw. Gebiete 21 20–44.
  • [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.
  • [7] Erickson, K. B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 263–291.
  • [8] Feller, W. (1949). Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67 98–119.
  • [9] Flajolet, P. and Sedgewick, R. (1995). Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theoret. Comput. Sci. 144 101–124.
  • [10] Gnedin, A. V. (2004). The Bernoulli sieve. Bernoulli 10 79–96.
  • [11] Gnedin, A. V. (2006). Constrained exchangeable partitions. Discrete Mathematics and Theoretical Computer Science Proc. Ser. A G 391–398.
  • [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.
  • [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.
  • [14] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479.
  • [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.
  • [17] Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 34 468–492.
  • [18] de Haan, L. and Resnick, S. I. (1979). Conjugate Π-variation and process inversion. Ann. Probab. 7 1028–1035.
  • [19] Heyde, C. C. (1967). A limit theorem for random walks with drift. J. Appl. Probab. 4 144–150.
  • [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.
  • [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.
  • [22] Kac, M. (1949). On deviations between theoretical and empirical distributions. Proc. Natl. Acad. Sci. USA 35 252–257.
  • [23] Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17 373–401.
  • [24] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
  • [25] Whittaker, E. T. and Watson, G. N. (1997). A Course in Modern Analysis. Cambridge Univ. Press, Cambridge.