Electronic Communications in Probability

Small deviations in $L_2$-norm for Gaussian dependent sequences

Seok Young Hong, Mikhail Lifshits, and Alexander Nazarov

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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 \[ \ln \mathbb{P} \left ( \sum _{k\in \mathbb{Z} } d_k^2 U_k^2 \le \varepsilon ^2\right ) \sim - M \varepsilon ^{-\frac{2} {2p-1}}, \qquad \textrm{ as } \varepsilon \to 0, \] whenever \[ d_k\sim d_{\pm } |k|^{-p}\quad \textrm{for some } p>\frac{1} {2} \, , \quad k\to \pm \infty , \] using the arguments based on the spectral theory of pseudo-differential operators by M. 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.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 41, 9 pp.

Received: 17 November 2015
Accepted: 10 May 2016
First available in Project Euclid: 19 May 2016

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx]

small deviations spectral asymptotics stationary sequences

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Hong, Seok Young; Lifshits, Mikhail; Nazarov, Alexander. Small deviations in $L_2$-norm for Gaussian dependent sequences. Electron. Commun. Probab. 21 (2016), paper no. 41, 9 pp. doi:10.1214/16-ECP4708. https://projecteuclid.org/euclid.ecp/1463663122

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