Small deviations in $L_2$-norm for Gaussian dependent 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 \[ \ln \mathbb{P} \left ( \sum _{k\in \mathbb{Z} } d_k^2 U_k^2 \le \varepsilon ^2\right ) \sim - M \varepsilon ^{-\frac{2} {2p-1}}, \qquad \textrm{ as } \varepsilon \to 0, \] 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.