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

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 ${\mathrm{\ell }}_2$-norm small deviation probabilities. It is shown that $$\ln \mathbb{P}(\sum _{k\in \mathbb{Z}}{d}_k^2{U}_k^2\le {\epsilon }^2)\sim -M{\epsilon }^{-\frac{2}{2p-1}},\text{ as }\epsilon \to 0,$$ whenever $$d_k\sim d_{\pm }\mathrm{\mid }k{\mathrm{\mid }}^{-p}\text{for some }p>\frac{1}{2},k\to \pm \mathrm{\infty },$$ 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.