Linear rigidity of stationary stochastic processes
2017 ◽
Vol 38
(7)
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pp. 2493-2507
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Keyword(s):
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.
2020 ◽
Vol 378
(2166)
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pp. 20190059
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1973 ◽
Vol 10
(04)
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pp. 881-885
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2015 ◽
Vol 52
(4)
◽
pp. 1003-1012
◽
2015 ◽
Vol 52
(04)
◽
pp. 1003-1012
◽