Small deviations of sums of correlated stationary Gaussian sequences

TL;DR

This paper investigates small deviation probabilities for sums of stationary Gaussian sequences, providing general results for various boundary behaviors and connecting discrete analogs of fractional Brownian motion to these probabilities.

Contribution

It extends the understanding of small deviation probabilities to sums of stationary Gaussian sequences, especially for boundaries tending to zero or infinity, and links discrete FBM to these probabilities.

Findings

Lower bounds for SDP transfer from FBM to discrete sequences

General results for SDP with constant and vanishing boundaries

Insights into spectral structure's role in upper bounds

Abstract

We consider the small deviation probabilities (SDP) for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogs of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well studied FBM caseto the discrete time setting under the usual assumptions that imply weak convergence while the transfer of the corresponding upper bounds necessarily requires a deeper knowledge of the spectral structure of the underlying stationary sequence.