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Penalisations of a Bessel process with dimension d(0 d 2) by a function of the ranked lengths of its excursions

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Penalising Brownian Paths

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1969))

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Abstract

For any integer n = 1, 2, ⋯, limiting laws, as t→∞, for a Bessel process with dimension d (0 < d < 2) penalised by the nth-ranked length of its excursions up to t, or up to the last zero before t, or again up to the first zero after t, are shown to exist, and are characterized.

Under these limiting laws Q (n), the canonical process admits a last zero g, and the sequence of the normalized ranked lengths of its excursions up to g, is described in terms of the Poisson-Dirichlet distribution studies, e.g., by Pitman-Yor [PY5]. As n → ∞, Q (n) is shown to converge to Q (∞) the distribution of the Bessel process penalised by an adequate function of its local time at 0, and the sequence of the normalized ranked lengths of the excursions up to g under Q (∞) is then precisely the Poisson-Dirichlet distribution PD1 − d/2 , 0 studied in Pitman-Yor [PY5].

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Authors and Affiliations

  1. Université Nancy I Institut Elie Cartan, Faculté des Sciences Departement Mathésmatiques , B.P. 239 54506 Vandoeuvre-les-Nancy, CX, France

    Bernard Roynette

  2. Université Paris VI Laboratoire de Probabilités, 175 rue de Chevaleret, 75013, Paris, France

    Marc Yor

Authors
  1. Bernard Roynette
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  2. Marc Yor
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Correspondence to Bernard Roynette .

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Roynette, B., Yor, M. (2009). Penalisations of a Bessel process with dimension d(0 d 2) by a function of the ranked lengths of its excursions. In: Penalising Brownian Paths. Lecture Notes in Mathematics(), vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89699-9_4

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