# On small deviations of stationary Gaussian processes and related
analytic inequalities

Research paper by **Michel J. G. Weber**

Indexed on: **27 Jul '11**Published on: **27 Jul '11**Published in: **Mathematics - Probability**

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#### Abstract

Let $ \{X_j, j\in \Z\}$ be a Gaussian stationary sequence having a spectral
function $F$ of infinite type. Then for all $n$ and $z\ge 0$,$$
\P\Big\{\sup_{j=1}^n |X_j|\le z \Big\}\le
\Big(\int_{-z/\sqrt{G(f)}}^{z/\sqrt{G(f)}} e^{-x^2/2}\frac{\dd x}{\sqrt{2\pi}}
\Big)^n,$$ 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), t\in \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 $T\ge \e(a)$, $$\P\Big\{\sup_{0\le s,t\le T} |X(s)-X(t)|\le
a\Big\} \le \exp \Big \{-{KT \over \e(a) p(\e(a))}\Big\}
,$$ where $\e (a)= \min\big\{b>0: \d (b)\ge a\big\}$, $\d (b)=\min_{u\ge
1}\{\sqrt{2(1-\g((ub))}, u\ge 1\}$, and
$ p(b) = 1+\sum_{j=2}^\infty {|2\g (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.