Linear rigidity of stationary stochastic processes
Résumé
We consider stationary stochastic processes $\{X_n : n \in Z\}$ such that $X_0$ lies in the closed linear span of $\{X_n : n = 0\}$; 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 vanish at zero and belong to the Zygmund class $\Gamma^*(1)$. We next give 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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