no. 2
A class of special subordinators with nested ranges
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 533-544.

Nous construisons, sur un unique espace de probabilités, une famille d’ensembles régénératifs ℛ (α) , indexée par toutes les fonctions mesurables α:[0,1]→[0,1]. Pour une fonction donnée α, l’ensemble ℛ (α) a même loi que l’image d’un subordinateur spécial. Les fonctions constantes correspondent aux subordinateurs stables. Si α≤β, on a ℛ (α) ⊂ℛ (β) . D’autres exemples de subordinateurs spéciaux sont donnés dans le cas discret.

We construct, on a single probability space, a class of regenerative sets ℛ (α) , indexed by all measurable functions α:[0,1]→[0,1]. For each function α, ℛ (α) , has the law of the range of a special subordinator. Constant functions correspond to stable subordinators. If α≤β, then ℛ (α) ⊂ℛ (β) . Other examples of special subordinators are given in the lattice case.

DOI : 10.1214/13-AIHP595
Classification : 60G51
Mots clés : regenerative set, subordinator, Bernstein function 
@article{AIHPB_2015__51_2_533_0,
     author = {Marchal, P.},
     title = {A class of special subordinators with nested ranges},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {533--544},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     doi = {10.1214/13-AIHP595},
     mrnumber = {3335014},
     zbl = {1329.60123},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP595/}
}
TY  - JOUR
AU  - Marchal, P.
TI  - A class of special subordinators with nested ranges
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 533
EP  - 544
VL  - 51
IS  - 2
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP595/
DO  - 10.1214/13-AIHP595
LA  - en
ID  - AIHPB_2015__51_2_533_0
ER  - 
%0 Journal Article
%A Marchal, P.
%T A class of special subordinators with nested ranges
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 533-544
%V 51
%N 2
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP595/
%R 10.1214/13-AIHP595
%G en
%F AIHPB_2015__51_2_533_0
Marchal, P. A class of special subordinators with nested ranges. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 533-544. doi : 10.1214/13-AIHP595. http://archive.numdam.org/articles/10.1214/13-AIHP595/

[1] J. Bertoin. Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997) 1–91. Lecture Notes in Math. 1717. Springer, Berlin, 1999. | MR | Zbl

[2] R. M. Blumenthal and R. K. Getoor. Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 (1961) 493–516. | MR | Zbl

[3] R. A. Doney. Fluctuation Theory for Lévy Processes. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005. Lecture Notes in Mathematics 1897. Springer, Berlin, 2007. | MR | Zbl

[4] P. J. Fitzsimmons, B. Fristedt and L. A. Shepp. The set of real numbers left uncovered by random covering intervals. Z. Wahrsch. Verw. Gebiete 70 (1985) 175–189. | DOI | MR | Zbl

[5] B. Mandelbrot. Renewal sets and random cutouts. Z. Wahrsch. Verw. Gebiete 22 (1972) 145–157. | DOI | MR | Zbl

[6] P. Marchal. Nested regenerative sets and their associated fragmentation process. In Mathematics and Computer Science III. Trends Math. 461–470. Birkhäuser, Basel, 2004. | MR | Zbl

[7] V. Rivero. On random sets connected to the partial records of Poisson point process. J. Theoret. Probab. 16 (1) (2003) 277–307. | MR | Zbl

[8] R. Schilling, R. Song and Z. Vondracek. Bernstein Functions. Theory and Applications. De Gruyter Studies in Mathematics 37. Walter de Gruyter, Berlin, 2010. | MR | Zbl

[9] L. A. Shepp. Covering the line with random intervals. Z. Wahrsch. Verw. Gebiete 22 (1972) 163–170. | DOI | MR | Zbl

[10] R. Song and Z. Vondracek. Some remarks on special subordinators. Rocky Mountain J. Math. 40 (1) (2010) 321–337. | MR | Zbl

Cité par Sources :