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- W4300511193 abstract "Let $ {X_j, jin Z}$ be a Gaussian stationary sequence having a spectral function $F$ of infinite type. Then for all $n$ and $zge 0$,$$ PBig{sup_{j=1}^n |X_j|le z Big}le Big(int_{-z/sqrt{G(f)}}^{z/sqrt{G(f)}} e^{-x^2/2}frac{dd x}{sqrt{2pi}} Big)^n,$$ where $ G(f)$ is the geometric mean of the Radon Nycodim derivative of the absolutely continuous part $f$ of $F$. The proof uses properties of finite Toeplitz forms. Let $ {X(t), tin R}$ be a sample continuous stationary Gaussian process with covariance function $g(u) $. We also show that there exists an absolute constant $K$ such that for all $T>0$, $a>0$ with $Tge e(a)$, $$PBig{sup_{0le s,tle T} |X(s)-X(t)|le aBig} le exp Big {-{KT over e(a) p(e(a))}Big} ,$$ where $e (a)= minbig{b>0: d (b)ge abig}$, $d (b)=min_{uge 1}{sqrt{2(1-g((ub))}, uge 1}$, and $ p(b) = 1+sum_{j=2}^infty {|2g (jb)-g ((j-1)b)-g ((j+1)b)| over 2(1-g(b))}$. The proof is based on some decoupling inequalities arising from Brascamp-Lieb inequality. Both approaches are developed and compared on examples. Several other related results are established." @default.
- W4300511193 created "2022-10-03" @default.
- W4300511193 creator A5041036701 @default.
- W4300511193 date "2011-04-14" @default.
- W4300511193 modified "2023-09-27" @default.
- W4300511193 title "On small deviations of stationary Gaussian processes and related analytic inequalities" @default.
- W4300511193 doi "https://doi.org/10.48550/arxiv.1104.2786" @default.
- W4300511193 hasPublicationYear "2011" @default.
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