# Are squared Bessel bridges infinitely divisible

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Are squared Bessel bridges infinitely divisible SÉMINAIRE DE PROBABILITÉS (STRASBOURG) NATHALIE EISENBAUM Are squared Bessel bridges infinitely divisible Séminaire de probabilités (Strasbourg), tome 35 (2001), p. 421-424. <http://www.numdam.org/item?id=SPS_2001__35__421_0> © Springer-Verlag, Berlin Heidelberg New York, 2001, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ARE SQUARED BESSEL BRIDGES INFINITELY DIVISIBLE ? NATHALIE EISENBAUM Laboratoire de Probabilités - Université Paris VI .~, Place Jussieu - Case 188 - 75252 Paris Cedex 05 Abstract: Consider a squared Bessel bridge between two positive values x and y. If x or y is equal to 0, then this process is infinitely divisible. In the case when both x and y are strictly positive, Pitman and Yor conjectured in [P-Y] that the process is not infinitely divisible. We show here that it is not infinitely decomposable in the sense of Shiga and Watanabe [S-W]. 1 - Introduction Let C be the canonical space C(IR+, IR+) and .~ be the ~-field, ~~W -~ w(s) = XS (c~); s > 0} . For d > 0 and x > 0, let of be the distribution on (C, 0) of the square of a Bessel process with dimension d starting from In [S-W], Shiga and Watanabe have established the following important additivity property: = (1) where , for P and Q two probabilities on (C, .~) , P (B Q denotes the distribution of 0) with (Xt, t > 0) and (Yt, t > 0) two independent processes respectively P and Q distributed. An immediate consequence of the above additivity property

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