Linear rigidity of stationary stochastic processes

TL;DR

This paper investigates the linear rigidity of stationary stochastic processes, establishing spectral conditions for rigidity, providing criteria for determinantal point processes, and demonstrating non-rigidity in a specific two-dimensional case.

Contribution

It introduces spectral criteria for linear rigidity, extends results to determinantal point processes, and shows a counterexample in higher dimensions.

Findings

Spectral density vanishing at zero implies rigidity.

Determinantal point processes on $ extbf{Z}$ and $ extbf{R}$ can be rigid under certain conditions.

The determinantal process on $ extbf{R}^2$ with Dyson sine-kernels is not rigid.

Abstract

We consider stationary stochastic processes $X_n$, $n\in \mathbb{Z}$ such that $X_0$ lies in the closed linear span of $X_n$, $n\neq 0$; 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 rigid, that the spectral density vanish at zero and belong to the Zygmund class $\Lambda_{*}(1)$. 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 $\textit{not}$ linearly rigid.