Small deviations in $L_2$-norm for Gaussian dependent sequences
ELECTRONIC COMMUNICATIONS IN PROBABILITY(2016)
摘要
Let U = (U-k)(k is an element of Z) be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted l(2)-norm small deviation probabilities. It is shown that ln P (Sigma(k is an element of Z) d(k)(2)U(k)(2) <= epsilon(2)) similar to M epsilon-2/2p-1, as epsilon -> 0, whenever d(k) similar to d(+/-)vertical bar k vertical bar(-p) for some p > 1/2, k -> +/-infinity, 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.
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关键词
small deviations,spectral asymptotics,stationary sequences
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