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Small deviations in \(L_2\)-norm for Gaussian dependent sequences. (English) Zbl 1336.60075
Summary: Let \(U=(U_k)_{k\in\mathbb Z}\) be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted \(\ell _2\)-norm small deviation probabilities. It is shown that \[ \ln\mathbb P \left(\sum\limits_{k\in\mathbb Z} d^2_k U^2_k \leqslant \varepsilon^2\right)\sim -M\varepsilon^{-\frac{2}{2p-1}},\quad \text{as }\varepsilon\to 0, \] whenever \[ d_k\sim d_\pm | k|^{-p}\text{ for some }p >\frac{1}{2},\quad k\to\pm\infty, \] using the arguments based on the spectral theory of pseudo-differential operators by M. Sh. Birman and M. Z. Solomyak [Vestn. Leningr. Univ., Math. 10, 237–247 (1982; Zbl 0492.47027); ibid. 12, 155–161 (1980; Zbl 0461.47025); Russ. Math. Surv. 32, No. 1, 15–89 (1977; Zbl 0376.47023)]. 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.

60G15 Gaussian processes
47G30 Pseudodifferential operators
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