The Annals of Probability

Stationary Processes Indexed by a Homogeneous Tree

Jean-Pierre Arnaud

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Abstract

Let $T$ be the set of vertices of a homogeneous tree and let $(X_t)_{t\in T}$ be a second-order real or complex-valued process such that the expected value $\mathbb{E}(X_s\bar{X}_t)$ depends only on the distance between the vertices $s$ and $t$. In this paper we construct a measure space $(K, \mathscr{H}, m)$ and an isometry of the closed subspace of $L^2_\mathbb{C}(\Omega, \mathscr{A}, P)$ spanned by $(X_t)_{t\in T}$ onto $L^2(m)$.

Article information

Source
Ann. Probab., Volume 22, Number 1 (1994), 195-218.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988856

Digital Object Identifier
doi:10.1214/aop/1176988856

Mathematical Reviews number (MathSciNet)
MR1258874

Zentralblatt MATH identifier
0793.60039

JSTOR
links.jstor.org

Subjects
Primary: 60G10: Stationary processes
Secondary: 60G15: Gaussian processes 60B99: None of the above, but in this section

Keywords
Stationary processes time series symmetric spaces Gelfand pairs homogeneous trees Cartier-Dunau polynomials

Citation

Arnaud, Jean-Pierre. Stationary Processes Indexed by a Homogeneous Tree. Ann. Probab. 22 (1994), no. 1, 195--218. doi:10.1214/aop/1176988856. https://projecteuclid.org/euclid.aop/1176988856


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