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Linear rigidity of stationary stochastic processes

  • ALEXANDER I. BUFETOV (a1) (a2) (a3) (a4), YOANN DABROWSKI (a5) and YANQI QIU (a1)

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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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