# $L_2$-Small Deviations for Weighted Stationary Processes

## TL;DR

This paper derives the logarithmic asymptotics of small deviation probabilities in the L2 norm for weighted stationary Gaussian processes, utilizing spectral theory of pseudo-differential operators.

## Contribution

It extends existing results by providing asymptotic estimates for a broad class of weighted stationary Gaussian processes using advanced spectral analysis techniques.

## Key findings

- Logarithmic asymptotics for small deviation probabilities derived
- Applicable to both real and complex-valued processes with power-type spectra
- Utilizes spectral theory of pseudo-differential operators

## Abstract

We find logarithmic asymptotics of $L_2$-small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having power-type discrete or continuous spectrum. As in the recent work by Hong, Lifshits and Nazarov, our results are based on the spectral theory of pseudo-differential operators developed by Birman and Solomyak.