Moment and Tail Bounds on the Wiener Chaos up to a Given Order

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Lebesgue spaces $L^{p}(\gamma_{c})$ and norms $\lVert\cdot\rVert_{p}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue and the chaos up to order $m$ $\mathcal{H}_{\le m}$ of The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space. $\exp$ is the exponential function, $\mathrm{e}=\exp(1)$, and powers of positive real numbers with real exponents are those of Real Power of a Positive Real Number. Let $m\in\mathbb{N}_{0}$ and $F\in\mathcal{H}_{\le m}$, and let $f:X\to\mathbb{R}$ be a Borel function in the class $F$.

1. (Moment bounds)¶ For every real $q\ge2$, $\int_{X}|f|^{q}\,d\gamma_{c}<\infty$, so that $F\in L^{q}(\gamma_{c})$, and

$$\lVert F\rVert_{q}\le(q-1)^{m/2}\,\lVert F\rVert_{2},$$

where $(q-1)^{0}=1$.

2. (Tail bound)¶ Suppose $m\ge1$, and let $\sigma\in\mathbb{R}$ be positive with $\lVert F\rVert_{2}\le\sigma$. Then for every real $s\ge(2\mathrm{e})^{m/2}\sigma$,

$$\gamma_{c}\bigl(\{x\in X:|f(x)|\ge s\}\bigr)\le\exp\Bigl(-\frac{m}{2\mathrm{e}}\Bigl(\frac{s}{\sigma}\Bigr)^{2/m}\Bigr).$$