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Probability Theory and Related Fields

, Volume 118, Issue 4, pp 439–454 | Cite as

The Markov chain associated to a Pick function

  • Gérard Letac
  • Dhafer Malouche

Abstract.

To each function ϕ˜(ω) mapping the upper complex half plane ?+ into itself such that the coefficient of ω in the Nevanlinna integral representation is one, we associate the kernel p(y, dx) of a Markov chain on ℝ by

$$$$

The aim of this paper is to study this chain in terms of the measure μ appearing in the Nevanlinna representation of ϕ˜(ω). We prove in particular three results. If x2 is integrable by μ, a law of large numbers is available. If μ is singular, i.e. if ϕ˜ is an inner function, then the operator P on L(ℝ) for the Lebesgue measure is the adjoint of T defined on L1(ℝ) by T(f)(ω) = f(ϕ(ω)), where ϕ is the restriction of ϕ˜ to ℝ. Finally, if μ is both singular and with compact support, we give a necessary and sufficient condition for recurrence of the chain.

Keywords

Markov Chain Lebesgue Measure Integral Representation Compact Support Half Plane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Gérard Letac
    • 1
  • Dhafer Malouche
    • 1
  1. 1.Laboratoire de Statistique et Probabilités, 118, Route de Narbonne, 31062 Toulouse Cedex, France. e-mail: letac@cict.frFR

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