×

Regenerative compositions in the case of slow variation. (English) Zbl 1114.60030

An increasing Lévy process (subordinator) \(S\) defines via its range a random division of \(\mathbb{R}_+\) into open interval components (gaps). An independent Poisson process \(\Pi_n\) with rate \(n e^{-x}, x> 0\), is considered for \(n\) large. The number \(K_n\) of gaps hit by at least one point of \(\Pi_n\) can be viewed as terminal value \(\mathcal{K} (\infty)\) of the process \(\mathcal{K}_n (T)\) counting the number of jumps of \(S\) such that \(\Pi_n \cap (S_{t^{-}}, S_t) \neq \emptyset\). The authors extend previous work on this subject and consider the random fluctuation of the process \(\mathcal{K}_n\) in the case when the tail of the Lévy measure of \(S\) is slowly varying. The motivation of this work comes from the study of regenerative composition structures.

MSC:

60G09 Exchangeability for stochastic processes
60C05 Combinatorial probability
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arratia, R.; Barbour, A. D.; Tavaré, S., (Logarithmic Combinatorial Structures: A Probabilistic Approach. Logarithmic Combinatorial Structures: A Probabilistic Approach, European Math. Soc. Monographs in Math., vol. 1 (2003)) · Zbl 1040.60001
[2] Bertoin, J., Lévy Processes (1996), Cambridge University Press · Zbl 0861.60003
[3] Bertoin, J., (Subordinators: Examples and Applications. Subordinators: Examples and Applications, Springer Lecture Notes in Math., vol. 1727 (1996)) · Zbl 0955.60046
[4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge University Press · Zbl 0617.26001
[5] Dudley, R. M., Probabilities and Metrics, (Lecture Notes Series, vol. 45 (1976), Aarhus Universitet) · Zbl 0355.60004
[6] Feller, W., An Introduction to Probability Theory and its Applications, vol. II (1971), Wiley · Zbl 0219.60003
[7] Gnedin, A. V., The Bernoulli sieve, Bernoulli, 10, 79-96 (2004) · Zbl 1044.60005
[8] Gnedin, A. V.; Pitman, J., Regenerative composition structures, Ann. Probab., 33, 445-479 (2005) · Zbl 1070.60034
[9] Gnedin, A. V.; Pitman, J., Regenerative partition structures, Electron. J. Combin., 11, 2 (2004/2005), paper #R12 · Zbl 1078.60009
[10] A.V. Gnedin, J. Pitman, M. Yor, Asymptotic laws for compositions derived from transformed subordinators, Ann. Probab. (2006) (in press). http://arxiv.org/abs/math.PR/0403438; A.V. Gnedin, J. Pitman, M. Yor, Asymptotic laws for compositions derived from transformed subordinators, Ann. Probab. (2006) (in press). http://arxiv.org/abs/math.PR/0403438 · Zbl 1142.60327
[11] Gnedin, A. V.; Pitman, J.; Yor, M., Asymptotic laws for regenerative compositions: gamma subordinators and the like, Probab. Theory Related Fields (2006) · Zbl 1099.60023
[12] Last, G.; Brandt, A., Marked Point Processes on the Real Line: The Dynamic Approach (1995), Springer: Springer NY · Zbl 0829.60038
[13] Müller, D. W., Verteilungs-Invarianzprinzipien für das starke Gesetz der großen Zahl, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 10, 173-192 (1968) · Zbl 0169.49801
[14] J. Pitman, Combinatorial Stochastic Processes, in: Springer Lecture Notes Math., 2005 (in press); J. Pitman, Combinatorial Stochastic Processes, in: Springer Lecture Notes Math., 2005 (in press)
[15] Winkel, M., Electronic foreign exchange markets and level passage events of multivariate subordinators, J. Appl. Probab., 42, 138-152 (2005) · Zbl 1078.60035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.