Linear rigidity of stationary stochastic processes

2015/07/02 by Bufetov, Alexander I., Dabrowski, Yoann, Qiu, Yanqi
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1507.00670

Abstract

We consider stationary stochastic processes X_n, n∈ ℤ such that X₀ 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 rigid, that the 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 on ℝ to be rigid. Finally, we show that the determinantal point process on ℝ² induced by a tensor square of Dyson sine-kernels is not linearly rigid.

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