Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann
[Infinitely divisible Wald's couples. Examples linked with the Euler gamma and the Riemann zeta functions]
Annales de l'Institut Fourier, Volume 55 (2005) no. 4, pp. 1219-1283.

To any positive measure c on + , such that : 0 (xx 2 )c(dx)< we associate an infinitely divisible Wald couple, i.e. : a couple of random variables (X,H) such that X and H are infinitely divisible, H0, and for any λ0,Ee λX ·Ee -λ 2 2H =1. More generally, to a positive measure c on + which satisfies : 0 e -αx x 2 c(dx)< for every α>α 0 , we associate an “Esscher family” of infinitely divisible Wald couples. We give many examples of such Esscher families and we prove that the particular ones which are associated with the gamma and the zeta functions enjoy remarkable properties.

A toute mesure c positive sur + telle que 0 (xx 2 )c(dx)<, nous associons un couple de Wald indéfiniment divisible, i.e. un couple de variables aléatoires (X,H) tel que X et H sont indéfiniment divisibles, H0, et pour tout λ0,Ee λX ·Ee -λ 2 2H =1. Plus généralement, à une mesure c positive sur + telle que 0 e -αx x 2 c(dx)< pour tout α>α 0 , nous associons une “famille d’Esscher” de couples de Wald indéfiniment divisibles. Nous donnons de nombreux exemples de telles familles d’Esscher. Celles liées à la fonction gamma et à la fonction zeta de Riemann possèdent des propriétés remarquables.

DOI: 10.5802/aif.2125
Classification: 60E67, 60E05, 60E10, 60G51
Mot clés : transformées de Laplace, lois indéfiniment divisibles, couples de Wald, fonctions gamma et zeta
Keywords: Laplace transforms, infinitely divisible laws, Wald couples, gamma and zeta functions
Roynette, Bernard 1; Yor, Marc 

1 Institut Elie Cartan, département de Mathématiques, B.P. 239, 54506 Vandoeuvre Les Nancy Cedex (France), Université Paris VI, Laboratoire de Probabilités et Modèles Aléatoires, Tour 56 - 3ème étage, 75252 Paris Cedex 05 (France)
     author = {Roynette, Bernard and Yor, Marc},
     title = {Couples de {Wald} ind\'efiniment divisibles. {Exemples} li\'es \`a la fonction gamma {d'Euler} et \`a la fonction zeta de {Riemann}},
     journal = {Annales de l'Institut Fourier},
     pages = {1219--1283},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
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     year = {2005},
     doi = {10.5802/aif.2125},
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Roynette, Bernard; Yor, Marc. Couples de Wald indéfiniment divisibles. Exemples liés à la fonction gamma d'Euler et à la fonction zeta de Riemann. Annales de l'Institut Fourier, Volume 55 (2005) no. 4, pp. 1219-1283. doi : 10.5802/aif.2125.

[BBE] P. Bougerol; M. Babillot; L. Elie The random difference equation X n =A n X n-1 +B n in the critical case, Ann. Prob., Volume 25 (1997), pp. 478-493 | DOI | MR | Zbl

[BFSS] C. Banderier ; P. Flajolet; G. Schaeffer ; M. Soria Random maps, coalescing saddles, singularity analysis and Airy phenomena, Random Struct. Algor., Volume 19 (2001), pp. 194-246 | DOI | MR | Zbl

[Bi] P. Biane La fonction zêta et les probabilités, La fonction zêta, Éditions de l'École Polytechnique, 2003 | MR

[BPY] P. Biane ; J. Pitman; M. Yor Probabilistic interpretation of the Jacobi theta and the Riemann zeta functions, via Brownian excursions, Bull. AMS, Volume 38 (2001), pp. 435-465 | MR | Zbl

[Br] M.F. Bru Wishart processes, J. Theor. Prob., Volume 4 (1991), pp. 725-751 | DOI | MR | Zbl

[BY] P. Biane; M. Yor Valeurs principales associées aux temps locaux browniens, Bull. Sci. Math., Volume 2 (1987) no. 111, pp. 23-101 | MR | Zbl

[Ca] R. Campbell Les intégrales eulériennes et leurs applications, Dunod, 1966 | MR | Zbl

[CY] L. Chaumont; M. Yor Exercises in Probability. A guided Tour from Measure Theory to Random Processes, via conditioning, Cambridge Series in Stat. and Prob. Math., 2003 | MR | Zbl

[DDMY] C. Donati-Martin; Y. Doumerc; H. Matsumoto; M. Yor Some properties of the Wishart processes and a matrix extension of the Hartman Watson laws, Publ. RIMS Kyoto Univ., Volume 40 (2004) no. 4, pp. 1385-1412 | DOI | MR | Zbl

[DGY] C. Donati-Martin; R. Ghomrasni; M. Yor Affine random equations and the stable 1 2 distribution, Stud. Sci. Math. Hung., Volume 36 (2000), pp. 347-405 | MR | Zbl

[DRVY] B. De Meyer; B. Roynette ; P. Vallois; M. Yor On independent times and positions for Brownian motions, Rev. Mat. Iberoamericana, Volume 18 (2002), pp. 541-586 | MR | Zbl

[Er] A. Erdelyi ; al. Higher transcendental Functions, I, Mc Graw Hill, 1953 | Zbl

[Go] L. Gordon A stochastic Approach to the Gamma Function, Amer. Math. Monthly, Volume 101 (1994), pp. 858-865 | DOI | MR | Zbl

[Gr] B. Grigelionis On the self decomposability of Euler's gamma function, trad. in Lituanian Math., Volume 43 (2003) no. 5, pp. 295-385 | MR | Zbl

[Ha] P. Hartman Completely monotone families of solutions of n-th order linear differential equations and infinitely divisible distributions, Ann. Scuola Normale Sup. Pisa, Volume IV-III (1976) no. 2, pp. 267-287 | Numdam | MR | Zbl

[JPY] M. Jeanblanc; J. Pitman; M. Yor Self similar processes with independent increments associated with Lévy and Bessel Processes, Stoch. Proc. Appl., Volume 100 (2002), pp. 188-223 | MR | Zbl

[JV] Z.J. Jurek; W. Vervaat An integral representation for self-decomposable Banach space valued random variables, Zeit. Wahr. Verw. Gebiet, Volume 62 (1983), pp. 247-262 | DOI | MR | Zbl

[Kh] A.Ya. Khintchine Limit for Sums of independent Random variables, Moscow and Lex, 1938

[Leb] N.N. Lebedev Special functions and their applications, Dover Pub. Inc., 1972 | MR | Zbl

[LeG] J.F. Le Gall Spatial Branching Processes, Random Snakes and Partial Differential Equations, Lecture Notes in Math., ETH Zürich, Birkhaüser, 1997 | MR | Zbl

[Let] G. Letac A characterization of the Gamma distribution, Adv. App. Prob., Volume 17 (1985), pp. 911-912 | DOI | MR | Zbl

[LH] G.D. Lin; C.Y. Hu The Riemann zeta distribution, Bernoulli, Volume 7 (2001), pp. 817-828 | DOI | MR | Zbl

[Lu1] E. Lukacs Characteristic functions, 2nd ed., Griffin, London, 1970 | MR | Zbl

[Lu2] E. Lukacs Contribution to a problem of D. Van Dantzig, Th. Prob. Appl., Volume XIII (1968) no. 1, pp. 116-127 | MR | Zbl

[MNY] H. Matsumoto; L. Nguyen-Ngoc; M. Yor; éd. H. Engelbert, R. Buckdahn Subordinators related to the exponential functionals of Brownian bridges and explicit formulae for the semigroups of hyperbolic Brownian motions, Proceedings École d'Hiver de Siegmundsburg, `Stochastic Processes and Related Topics' (2000)

[MSW] M. Maejima; K. Sato; T. Watanabe Completely operator self-decomposable distributions, Tokyo J. Math., Volume 23 (2000), pp. 235-253 | DOI | MR | Zbl

[Mu] R.J. Muirhead Aspects of Multivariate Statistical Theory, Wiley Series in Prob. and Math. Stat., 1982 | MR | Zbl

[Pa] S.J. Patterson An introduction to the theory of the Riemann Zeta Function, Cambridge University Press, 1988 | MR | Zbl

[Ri] B. Riemann Über die Anzahl der Primzahlen unter eine gegebner Grösse, Monatsber. Akad. Berlin (1859), pp. 671-680

[RVY] B. Roynette; P. Vallois; M. Yor Limiting laws associated with brownian motion perturbed by normalized exponential weights (2005) Studia Math. Hung. (à paraître) | Zbl

[RY] D. Revuz; M. Yor Continuous Martingales and Brownian Motion, Gründ. der Math. Wissenschaft, 3e éd., Springer Verlag, Basel, 1999 | MR | Zbl

[Sa1] K. Sato Lévy processes and infinitely divisible distributions, Cambridge Univ. Press., 1999 | MR | Zbl

[Sa2] K. Sato Self similar processes with independent increments, Prob. Th. Rel. Fields, Volume 89 (1991), pp. 285-300 | DOI | MR | Zbl

[Sch] I.J. Schoenberg On Polya Frequency Functions I, J. Anal. Math., Volume 1 (1951), pp. 331-374 | DOI | MR | Zbl

[Va] G. Valiron Théorie des fonctions, 2e éd., Masson, 1955 | Zbl

[WW] E.T. Whittaker; G.N. Watson A course of Modern Analysis, 4e éd., Cambridge University Press, 1927 | JFM | MR

[Yo1] M. Yor Some aspects of Brownian Motion II, Some recent Martingales problems, Lecture Notes in Math., ETZ Zürich, Birkhaüser, 1997 | MR | Zbl

[Yo2] M. Yor Exponential functionals of Brownian Motion and Related Processes, Springer Finance, 2001 | MR | Zbl

[Yo3] M. Yor Loi de l'indice du lacet brownien et distribution de Hartman Watson, Zeitschrift für Wahr. und Verw. Gebiete, Volume 53 (1980), pp. 71-95 | DOI | MR | Zbl

[Zo] V.M. Zolotarev One dimensional Stable Distributions, Translations of Math. Monographs, 65, Amer. Math. Soc., 1986 | MR | Zbl

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