This article is cited in 3 papers

Small deviations of sums of correlated stationary Gaussian sequences

F. Aurzada^a, M. A. Lifshits<sup>b

a</sup> Technische Universität Darmstadt
^b St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: 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.

Keywords: fractional Brownian motion, fractional Gaussian noise, Gaussian process, small deviation probability, stationary Gaussian sequence.

Received: 26.07.2016

DOI: 10.4213/tvp5081

 English version: Theory of Probability and its Applications, 2017, 61:4, 540–568

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