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.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 24 April 1998 / Revised version: 13 March 2000 / Published online: 20 October 2000
Rights 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
Issue Date:
DOI: https://doi.org/10.1007/PL00008750