N ov 2 01 6 Linear rigidity of stationary stochastic processes

We consider stationary stochastic processes {Xn : n ∈ Z} such thatX0 lies in the closed linear span of {Xn : n 6= 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, th at e spectral density vanish at zero and belong to the Zygmund class Λ∗(1). We next give sufficient condition for stationary determinantal point processes on Z and onR to be linearly rigid. Finally, we show that the determinantal point process on R2 induced by a tensor square of Dyson sine-kernels is not linearly rigid.