Small Deviations of Sums of Correlated Stationary Gaussian Sequences

We consider the small deviation probabilities (SDP) in the uniform norm 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 analogues of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well-studied FBM case to 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.