Semi-asymptotic bounds for moderate deviations of suprema of Gaussian polynomials

Abstract: We use decoupling inequalities for general Gaussian vectors and classical tools from Gaussian processes theory to study moderate deviations of suprema of trigonometric and almost periodic Gaussian polynomials. Let \begin{eqnarray*} X_x(t) &=& \sum_{k\le x} a_k \,\big( g_k \cos 2\pi k \,t+ g'k\sin 2\pi k \, t\, \big), \quad 0\le t\le 1, \ x>0, \end{eqnarray*} where $(g_k){k\ge 1}$, $(g'k){k\ge 1}$ are two independent sequences of i.i.d. $\mathcal N(0,1)$ distributed random variables, and $(a_k){k\ge 1}$ are real numbers such that $A_x = \sum{k\le x} a_k^2\uparrow \infty$ with $x$. Assume for instance that $A(x)\sim \log\log x$, $x\to \infty$ and $B=\sum_{k\ge 1} a_k^4<\infty$. Let $0<\eta< 1$. We prove that there exists an absolute constant $C$, such that for all $x$ large enough, \begin{align*} \P\Big{ \sup_{0\le t\le 1} X_x(t) \le\sqrt{2\eta (\log\log x)(\log \log\log x)}\Big} \ \le \ e^{-\,\frac{C (\log\log x)^{1-\eta}}{ \sqrt{8\eta (B+1)(\log \log\log x)}}}. \end{align*} We also establish an approximation theorem of moderate deviations of almost periodic Gaussian polynomials by Gaussian polynomials with $\Q$-frequencies, and show that the error term has an exponential decay. We finally study for general non-vanishing coefficient sequences, the behavior along lattices of almost periodic Gaussian polynomials with linearly independent frequencies.