## Asymptotic laws for regenerative compositions: gamma subordinators and the like.(English)Zbl 1099.60023

The random exponentially transfomed set in the interval $$[0,1]$$ generated by a Gamma subordinator generates clusters for $$n$$ uniform random points in $$[0,1]$$. The authors prove a central limit theorem for the number $$K_n$$ of components. The proof is based on the contraction method for the degenerate case of R. Neininger and the reviewer [Ann. Probab. 32, No. 3B, 2838–2856 (2004; Zbl 1060.60005)]. The paper complements previous work of the authors for the case that the tail of the Lévy measure is regularly varying at $$0$$.

### MSC:

 60G09 Exchangeability for stochastic processes 60C05 Combinatorial probability

Zbl 1060.60005
Full Text:

### References:

 [1] Armstrong, SIAM J. Math. Anal., 11, 300 (1980) · Zbl 0433.41016 [2] Barbour, A.D., Gnedin, A.V.: Regenerative compositions in the case of slow variation. 2005. (available at arXiv:math.PR/0505171) · Zbl 1114.60030 [3] Arratia, R., Barbour, A.D., Tavaré, S.: Logarithmic combinatorial structures: a probabilistic approach. European Math. Soc. Monographs in Math., v. 1, 2003 · Zbl 1040.60001 [4] Berg, C.; Duran, A. J., Some transformations of Hausdorff moment sequences and harmonic numbers, Ark. Math., 42, 239-257 (2004) · Zbl 1057.44002 [5] Fedoryuk, M.V.: Asymptotics: Integrals and series. Nauka, Moscow, 1987 · Zbl 0641.41001 [6] Feller, W.: An Introduction to Probability Theory and its Applications. volume II. Wiley, 2nd edition, 1971 · Zbl 0219.60003 [7] Flajolet, Theoret. Comput. Sci., 144, 3 (1995) · Zbl 0869.68057 [8] Gnedin, Ann. Probab., 25, 1437 (1997) · Zbl 0895.60037 [9] Gnedin, Bernoulli, 10, 79 (2004) · Zbl 1044.60005 [10] Gnedin, Combinatorics, Probab. Comput., 13, 185 (2004) · Zbl 1060.62039 [11] Gnedin, Ann. Probab., 33, 445 (2005) · Zbl 1070.60034 [12] Gnedin, A.V., Pitman, J.: Regenerative partition structures. Electron. J. Combin. 11, Research Paper 12, 21 pp. 2004/05 · Zbl 1078.60009 [13] Gnedin, A.V., Pitman, J., Yor, M.: Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 2005 (to appear, available at arXiv:math. PR/0403438) · Zbl 1142.60327 [14] Karlin, J. Math. Mech., 17, 373 (1967) · Zbl 0154.43701 [15] Neininger, Ann. Prob., 32, 2838 (2004) · Zbl 1060.60005 [16] Pitman, J.: Combinatorial stochastic processes. Lecture notes for St. Flour course (July 2002), Springer Lecture Notes Math. 2005. (available via http://www.stat.berkeley.edu) [17] Prodinger, H.: Compositions and patricia tries: no fluctuations in the variance! Proc. 6th Workshop on ALENEX and 1st Workshop on ANALCO, L. Arge et al (eds), SIAM, pp. 211-215 [18] Pitman, Canad. J. Math., 55, 292 (581) [19] Titchmarsh, E.C.: Introduction to the theory of Fourier integrals. Oxford Univ. Press 1937 · JFM 63.0367.05
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