Small Deviations in $L_2$-norm for Gaussian Dependent Sequences

TL;DR

This paper investigates the asymptotic behavior of small deviation probabilities in weighted $\ell_2$-norms for Gaussian stationary sequences, revealing how dependence affects these probabilities compared to independent cases.

Contribution

It introduces a spectral theory-based approach to analyze small deviations in Gaussian dependent sequences, highlighting the influence of dependence on the asymptotic constants.

Findings

Asymptotic probability decay rate characterized by $\varepsilon^{-rac{2}{2p-1}}$

Dependence structure influences the constant $M$ in the asymptotic expression

Method extends spectral theory techniques to dependent Gaussian sequences

Abstract

Let $U=(U_k)_{k\in\mathbb{Z}}$ be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted $\ell_2$-norm small deviation probabilities. It is shown that \[ \ln \mathbb{P}\left( \sum_{k\in\mathbb{Z}} d_k^2 U_k^2 \leq \varepsilon^2\right) \sim - M \varepsilon^{-\frac{2}{2p-1}}, \qquad \textrm{ as } \varepsilon\to 0, \] whenever \[ d_k\sim d_{\pm} |k|^{-p}\quad \textrm{for some } p>\frac{1}{2} \, , \quad k\to \pm\infty, \] using the arguments based on the spectral theory of pseudo-differential operators by M. Birman and M. Solomyak. The constant $M$ reflects the dependence structure of $U$ in a non-trivial way, and marks the difference with the well-studied case of the i.i.d. sequences.