Regenerative compositions in the case of slow variation: A renewal theory approach

Alexander Gnedin (Queen Mary University of London)
Alexander Iksanov (National T. Shevchenko University of Kiev)


A regenerative composition structure is a sequence of ordered partitions derived from the range of a subordinator by a natural sampling procedure. In this paper, we extend previous studies on the asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in the case when the Lévy measure of the subordinator has a property of slow variation at $0$. Using tools from the renewal theory the limit laws for $K_n$ are obtained in terms of integrals involving the Brownian motion or stable processes. In other words, the limit laws are either normal or other stable distributions, depending on the behavior of the tail of Lévy measure at $\infty$. Similar results are also derived for the number of singleton blocks.

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Pages: 1-19

Publication Date: September 17, 2012

DOI: 10.1214/EJP.v17-2002


  • Barbour, A. D.; Gnedin, A. V. Regenerative compositions in the case of slow variation. Stochastic Process. Appl. 116 (2006), no. 7, 1012--1047. MR2238612
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396
  • Bingham, N. H. Limit theorems for regenerative phenomena, recurrent events and renewal theory. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21 (1972), 20--44. MR0353459
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1989. xx+494 pp. ISBN: 0-521-37943-1 MR1015093
  • Durrett, Richard; Liggett, Thomas M. Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 275--301. MR0716487
  • Gnedin, Alexander V. The Bernoulli sieve. Bernoulli 10 (2004), no. 1, 79--96. MR2044594
  • Gnedin, Alexander V. Regeneration in random combinatorial structures. Probab. Surv. 7 (2010), 105--156. MR2684164
  • Gnedin, Alexander; Iksanov, Alexander; Marynych, Alexander. Limit theorems for the number of occupied boxes in the Bernoulli sieve. Theory Stoch. Process. 16 (2010), no. 2, 44--57. MR2777900
  • Gnedin, Alexander; Iksanov, Alexander; Marynych, Alexander. The Bernoulli sieve: an overview. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 329--341, Discrete Math. Theor. Comput. Sci. Proc., AM, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010. MR2735350
  • Gnedin, Alexander V.; Iksanov, Alexander M.; Negadajlov, Pavlo; Rösler, Uwe. The Bernoulli sieve revisited. Ann. Appl. Probab. 19 (2009), no. 4, 1634--1655. MR2538083
  • Gnedin, Alexander; Pitman, Jim. Regenerative composition structures. Ann. Probab. 33 (2005), no. 2, 445--479. MR2122798
  • Gnedin, Alexander; Pitman, Jim; Yor, Marc. Asymptotic laws for regenerative compositions: gamma subordinators and the like. Probab. Theory Related Fields 135 (2006), no. 4, 576--602. MR2240701
  • Gnedin, Alexander; Pitman, Jim; Yor, Marc. Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 34 (2006), no. 2, 468--492. MR2223948
  • Iksanov, Alexander. On the number of empty boxes in the Bernoulli sieve II. Stochastic Process. Appl. 122 (2012), no. 7, 2701--2729. MR2926172
  • Iksanov, A.: Functional limit theorems for renewal shot noise processes, ARXIV1202.1950
  • Iksanov, A., Marynych, A. and Meiners, M.: Moment convergence in renewal theory, ARXIV1208.3964
  • Sgibnev, M. S. On the renewal theorem in the case of infinite variance. (Russian) Sibirsk. Mat. Zh. 22 (1981), no. 5, 178--189, 224. MR0632826
  • Walker, Stephen; Muliere, Pietro. A characterization of a neutral to the right prior via an extension of Johnson's sufficientness postulate. Ann. Statist. 27 (1999), no. 2, 589--599. MR1714716
  • Whitt, Ward. Stochastic-process limits. An introduction to stochastic-process limits and their application to queues. Springer Series in Operations Research. Springer-Verlag, New York, 2002. xxiv+602 pp. ISBN: 0-387-95358-2 MR1876437

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