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Linear rigidity of stationary stochastic processes

Published online by Cambridge University Press:  03 April 2017

ALEXANDER I. BUFETOV,
YOANN DABROWSKI and
YANQI QIUOpen the ORCID record for YANQI QIU[Opens in a new window]
ALEXANDER I. BUFETOV
Affiliation:
Aix-Marseille Université, Centrale Marseille, CNRS, I2M, UMR7373, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France email yqi.qiu@gmail.comSteklov Institute of Mathematics, 8 Gubkina Street, Moscow, 119991, RussiaInstitute for Information Transmission Problems, Bolshoy Karetny per. 19, 127994, Moscow, RussiaNational Research University Higher School of Economics, Myasnitskaya ul., 20, Moscow, 101000, Russia email bufetov@mi.ras.ru
YOANN DABROWSKI
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 bd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France email dabrowski@math.univ-lyon1.fr
YANQI QIU
Affiliation:
Aix-Marseille Université, Centrale Marseille, CNRS, I2M, UMR7373, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France email yqi.qiu@gmail.com
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Abstract

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We consider stationary stochastic processes $\{X_{n}:n\in \mathbb{Z}\}$ such that $X_{0}$ lies in the closed linear span of $\{X_{n}:n\neq 0\}$; following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be linearly rigid, that the spectral density vanishes at zero and belongs to the Zygmund class $\unicode[STIX]{x1D6EC}_{\ast }(1)$. We next give a sufficient condition for stationary determinantal point processes on $\mathbb{Z}$ and on $\mathbb{R}$ to be linearly rigid. Finally, we show that the determinantal point process on $\mathbb{R}^{2}$ induced by a tensor square of Dyson sine kernels is not linearly rigid.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 7 , October 2018, pp. 2493 - 2507
Copyright
© Cambridge University Press, 2017 

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