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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.
