Linear rigidity of stationary stochastic processes

Abstract

We consider stationary stochastic processes $X_n$, $n\in \mathbb{Z}$ such that $X_0$ lies in the closed linear span of $X_n$, $n\ne 0$; 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 rigid, that the spectral density vanish at zero and belong to the Zygmund class ${\Lambda }_{\ast }(1)$. We next give sufficient condition for stationary determinantal point processes on $\mathbb{Z}$ and on $\mathbb{R}$ to be rigid. Finally, we show that the determinantal point process on ${\mathbb{R}}^2$ induced by a tensor square of Dyson sine-kernels is $\text{not}$ linearly rigid.

18 pp, to appear in Ergodic Theory and Dynamical Systems