## Bernoulli

• Bernoulli
• Volume 19, Number 4 (2013), 1350-1377.

### On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator

#### Abstract

Results about the laws of the perpetuity and remainder variables associated to a subordinator are presented, with particular emphasis on their Mellin transforms, and multiplicative infinite divisibility property. Previous results by Bertoin–Yor (Electron. Commun. Probab. 6 (2001) 95–106) are incorporated in our discussion; important examples when the subordinator is the inverse local time of a diffusion are exhibited. Results of Urbanik (Probab. Math. Statist. 15 (1995) 493–513) are also discussed in detail; they appear to be too little known, despite the fact that quite a few of them have priority upon other works in this area.

#### Article information

Source
Bernoulli, Volume 19, Number 4 (2013), 1350-1377.

Dates
First available in Project Euclid: 27 August 2013

https://projecteuclid.org/euclid.bj/1377612855

Digital Object Identifier
doi:10.3150/12-BEJSP01

Mathematical Reviews number (MathSciNet)
MR3102555

Zentralblatt MATH identifier
1287.60096

#### Citation

Hirsch, Francis; Yor, Marc. On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator. Bernoulli 19 (2013), no. 4, 1350--1377. doi:10.3150/12-BEJSP01. https://projecteuclid.org/euclid.bj/1377612855

#### References

• [1] Andrews, G.E., Askey, R. and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and Its Applications 71. Cambridge: Cambridge Univ. Press.
• [2] Artin, E. (1964). The Gamma Function. Athena Series: Selected Topics in Mathematics. New York: Holt, Rinehart and Winston. Translated by Michael Butler.
• [3] Berg, C. (1979). The Stieltjes cone is logarithmically convex. In Complex Analysis Joensuu 1978 (Proc. Colloq., Univ. Joensuu, Joensuu, 1978). Lecture Notes in Math. 747 46–54. Berlin: Springer.
• [4] Berg, C. (2005). On powers of Stieltjes moment sequences. I. J. Theoret. Probab. 18 871–889.
• [5] Berg, C. (2007). On powers of Stieltjes moment sequences. II. J. Comput. Appl. Math. 199 23–38.
• [6] Berg, C. and Durán, A.J. (2004). A transformation from Hausdorff to Stieltjes moment sequences. Ark. Mat. 42 239–257.
• [7] Bertoin, J. (1989). Applications de la théorie spectrale des cordes vibrantes aux fonctionnelles additives principales d’un brownien réfléchi. Ann. Inst. Henri Poincaré Probab. Stat. 25 307–323.
• [8] Bertoin, J. (1999). Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint–Flour, 1997). Lecture Notes in Math. 1717 1–91. Berlin: Springer.
• [9] Bertoin, J., Biane, P. and Yor, M. (2004). Poissonian exponential functionals, $q$-series, $q$-integrals, and the moment problem for log-normal distributions. In Seminar on Stochastic Analysis, Random Fields and Applications IV (R.C. Dalang, M. Dozzi and F. Russo, eds.). Progress in Probability 58 45–56. Basel: Birkhäuser.
• [10] Bertoin, J. and Yor, M. (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Commun. Probab. 6 95–106 (electronic).
• [11] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probab. Surv. 2 191–212.
• [12] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion (M. Yor, ed.). Bibl. Rev. Mat. Iberoamericana 73–130. Madrid: Rev. Mat. Iberoamericana.
• [13] Comtet, A., Texier, C. and Tourigny, Y. (2011). Supersymmetric quantum mechanics with Lévy disorder in one dimension. J. Stat. Phys. 145 1291–1323.
• [14] Donati-Martin, C. and Yor, M. (2006). Some explicit Krein representations of certain subordinators, including the gamma process. Publ. Res. Inst. Math. Sci. 42 879–895.
• [15] Donati-Martin, C. and Yor, M. (2007). Further examples of explicit Krein representations of certain subordinators. Publ. Res. Inst. Math. Sci. 43 315–328.
• [16] Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications 35. Cambridge: Cambridge Univ. Press.
• [17] Hirsch, F. (1975). Familles d’opérateurs potentiels. Ann. Inst. Fourier (Grenoble) 25 xxii, 263–288.
• [18] Hirsch, F. and Yor, M. (2013). On the remarkable Lamperti representation of the inverse local time of a radial Ornstein–Uhlenbeck process. Bull. Belgian Math. Soc. To appear.
• [19] Itô, M. (1973). Sur une famille sous-ordonnée au noyau de convolution de Hunt donné. Nagoya Math. J. 51 45–56.
• [20] Knight, F.B. (1981). Characterization of the Lévy measures of inverse local times of gap diffusion. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981) (E. Cinlar, K.L. Chung and R.K. Getoor, eds.). Progr. Prob. Statist. 1 53–78. Mass.: Birkhäuser.
• [21] Kotani, S. and Watanabe, S. (1982). Krein’s spectral theory of strings and generalized diffusion processes. In Functional Analysis in Markov Processes 1981, (M. Fukushima, ed.). Lecture Notes in Math. 923 235–259. Berlin: Springer.
• [22] Küchler, U. (1986). On sojourn times, excursions and spectral measures connected with quasidiffusions. J. Math. Kyoto Univ. 26 403–421.
• [23] Küchler, U. and Salminen, P. (1989). On spectral measures of strings and excursions of quasi diffusions. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 490–502. Berlin: Springer.
• [24] Molchanov, S.A. and Ostrovskiĭ, E. (1969). Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 14 128–131.
• [25] Pitman, J. and Yor, M. (1997). On the lengths of excursions of some Markov processes. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 272–286. Berlin: Springer.
• [26] Schilling, R.L., Song, R. and Vondraček, Z. (2010). Bernstein Functions: Theory and Applications. de Gruyter Studies in Mathematics 37. Berlin: de Gruyter.
• [27] Urbanik, K. (1995). Infinite divisibility of some functionals on stochastic processes. Probab. Math. Statist. 15 493–513.