Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
The Markov chain associated to a Pick function
Download PDF
Download PDF
  • Published: December 2000

The Markov chain associated to a Pick function

  • Gérard Letac1 &
  • Dhafer Malouche1 

Probability Theory and Related Fields volume 118, pages 439–454 (2000)Cite this article

  • 101 Accesses

  • 5 Citations

  • Metrics details

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 x 2 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 L 1(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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Laboratoire de Statistique et Probabilités, 118, Route de Narbonne, 31062 Toulouse Cedex, France. e-mail: letac@cict.fr, , , , , , FR

    Gérard Letac & Dhafer Malouche

Authors
  1. Gérard Letac
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dhafer Malouche
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 24 April 1998 / Revised version: 13 March 2000 / Published online: 20 October 2000

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Letac, G., Malouche, D. The Markov chain associated to a Pick function. Probab Theory Relat Fields 118, 439–454 (2000). https://doi.org/10.1007/PL00008750

Download citation

  • Issue Date: December 2000

  • DOI: https://doi.org/10.1007/PL00008750

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Markov Chain
  • Lebesgue Measure
  • Integral Representation
  • Compact Support
  • Half Plane
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature