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

Each result cited is universally quantified over the data in its own statement.

Powers of nonnegative reals with positive exponents are those of Real Power of a Nonnegative Real Number §power, which agree with Real Power of a Positive Real Number for positive bases, and their properties are those of Properties of Real Powers of Nonnegative Real Numbers. For real $r\ge1$ and a Borel $u$ with $\int_{X}|u|^{r}\,d\gamma_{c}<\infty$, $\lVert u\rVert_{r}=(\int_{X}|u|^{r}\,d\gamma_{c})^{1/r}$ by The Lebesgue Space of Power-Integrable Functions §norm and Power-Integrable Functions and the p-Seminorm §seminorm, and $\lVert u\rVert_{r}^{r}=\int_{X}|u|^{r}\,d\gamma_{c}$ by Elementary Properties of the p-Seminorm §power. We use that $\exp$ is positive and strictly increasing with $\exp(u+v)=\exp(u)\exp(v)$ and $\exp(-u)=1/\exp(u)$ (Basic Properties of the Exponential Function), so that $\log$ is strictly increasing on $(0,\infty)$, $\log1=0$ and $\log\exp(u)=u$ (The Natural Logarithm).

Step 1 (The setting with weights $c$). By A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian, $c$ is a variance sequence: its terms are positive and $\bar{c}=\sum_{k=1}^{\infty}c_{k}$ converges. For each $k$, the $k$-th partial sum is at least $c_{k}$, the other terms being positive, so $c_{k}\le\bar{c}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. Hence $c$ is a weight sequence, and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation may be instantiated with $c$ as the noise weights and $\bar{c}$ as the bound of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights. The objects in the present statement, namely $X$, $c$, $\gamma_{c}$, the spaces $L^{p}(\gamma_{c})$, the polynomials $H_{\alpha}$ and $\mathcal{H}_{\le m}$, do not involve the noise weights, so they are the same in this instance. Let $P^{c}_{r}$ ($r\ge0$) denote the Mehler semigroup with noise weights $c$. In this instance the rates of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates are $\theta_{k}=c_{k}/c_{k}=1$, so the weighted order of $\alpha\in\mathcal{A}$ is $\theta\cdot\alpha=\sum_{k=1}^{n}\alpha_{k}=|\alpha|$ for a length bound $n$ of $\alpha$, by Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order; and $\kappa=1$ satisfies $c_{k}\le\kappa\,c_{k}$ for every $k$.

Step 2 (Finite Hermite combinations). Let $q\ge2$ be real, let $S$ be a finite set of multi-indices $\alpha$ with $|\alpha|\le m$, let $b_{\alpha}\in\mathbb{R}$ for $\alpha\in S$, and let $u=\sum_{\alpha\in S}b_{\alpha}H_{\alpha}$ (the zero function if $S$ is empty). We show $\int_{X}|u|^{q}\,d\gamma_{c}<\infty$ and

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

Put $\tau=\frac{1}{2}\log(q-1)$; since $q-1\ge1$, $\tau\ge0$. Let $v=\sum_{\alpha\in S}b_{\alpha}\exp(\tau|\alpha|)H_{\alpha}$. By The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §class, applied in the instance of Step 1, $u$ and $v$ lie in $\mathcal{F}C_{\mathrm{pol}}(X)$ and in $L^{r}(\gamma_{c})$ for every real $r\ge1$. By the linearity of $P^{c}_{\tau}$ (The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §cylindrical) and The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §hermite with $\theta\cdot\alpha=|\alpha|$,

$$P^{c}_{\tau}v=\sum_{\alpha\in S}b_{\alpha}\exp(\tau|\alpha|)\exp(-\tau|\alpha|)H_{\alpha}=u .$$

Apply Hypercontractivity of the Mehler Semigroup with Noise Weights (Nelson's Theorem) §hypercontractive in the instance of Step 1, with $\kappa=1$, with $2$ and $q$ in place of $p$ and $q$, and with $t=\tau$: here $1<2\le q$, $0\le\tau$, and $\exp(2\tau)(2-1)=\exp(\log(q-1))=q-1$. It gives $\lVert u\rVert_{q}=\lVert P^{c}_{\tau}v\rVert_{q}\le\lVert v\rVert_{2}$. By The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product the norm $\lVert\cdot\rVert_{2}$ is that of the inner product $\langle\cdot,\cdot\rangle_{L^{2}(\gamma_{c})}$, so bilinearity and Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §orthogonality give

$$\lVert v\rVert_{2}^{2}=\sum_{\alpha\in S}b_{\alpha}^{2}\exp(2\tau|\alpha|)\,\alpha!\,c^{\alpha}\le\exp(2\tau m)\sum_{\alpha\in S}b_{\alpha}^{2}\,\alpha!\,c^{\alpha}=\exp(2\tau m)\,\lVert u\rVert_{2}^{2},$$

because $\exp(2\tau|\alpha|)\le\exp(2\tau m)$ for $|\alpha|\le m$ and $\tau\ge0$, and $\alpha!\,c^{\alpha}=\lVert H_{\alpha}\rVert_{2}^{2}\ge0$. As $\exp(2\tau m)=\exp(\tau m)^{2}$, we get $\lVert v\rVert_{2}\le\exp(\tau m)\lVert u\rVert_{2}$, and $\exp(\tau m)=\exp(\frac{m}{2}\log(q-1))=(q-1)^{m/2}$ by Real Power of a Positive Real Number, which equals $1=(q-1)^{0}$ when $m=0$. This proves (1).

Step 3 (Claim 1). Let $q\ge2$ be real. By The Wiener Chaoses of a Diagonal Gaussian Measure on a Hilbert Space §chaos-up-to, $F$ lies in the closure in $L^{2}(\gamma_{c})$ of the linear span $V$ of $\{H_{\alpha}:\alpha\in\mathcal{A},\ |\alpha|\le m\}$, and the metric of $L^{2}(\gamma_{c})$ is the distance of its norm $\lVert\cdot\rVert_{2}$ (The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product); so by Sequential Characterization of the Closure in a Metric Space there is a sequence $(F_{j})_{j\in\mathbb{N}}$ in $V$ with $\lVert F_{j}-F\rVert_{2}\to0$. Each $F_{j}$ is the zero vector or a finite sum $\sum_{l}t_{l}H_{\alpha_{l}}$ with $|\alpha_{l}|\le m$; collecting equal indices, $F_{j}$ is the class of a function $u_{j}$ as in Step 2, and by (1), $\lVert u_{j}\rVert_{q}\le(q-1)^{m/2}\lVert F_{j}\rVert_{2}$. By The Lebesgue Space of Power-Integrable Functions §space, $u_{j}-f$ is a representative of $F_{j}-F$, so $\int_{X}|u_{j}-f|^{2}\,d\gamma_{c}=\lVert F_{j}-F\rVert_{2}^{2}\to0$. Choose indices $j_{1}<j_{2}<\cdots$ with $\lVert F_{j_{i}}-F\rVert_{2}\le4^{-i}$, and let $A_{i}=\{x\in X:4^{-i}\le|u_{j_{i}}(x)-f(x)|^{2}\}$. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §markov, $A_{i}$ is Borel and $4^{-i}\gamma_{c}(A_{i})\le\lVert F_{j_{i}}-F\rVert_{2}^{2}\le16^{-i}$, so $\gamma_{c}(A_{i})\le(1/4)^{i}$. The series $\sum_{i}(1/4)^{i}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, so, as $0\le\gamma_{c}(A_{i})\le(1/4)^{i}$ for every $i$, the series $\sum_{i}\gamma_{c}(A_{i})$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, that is, $\sum_{i}\gamma_{c}(A_{i})<\infty$. Hence the first Borel-Cantelli lemma of Borel-Cantelli Lemmas, applied on the probability space $(X,\mathcal{B}(X),\gamma_{c})$ to the events $A_{i}$, gives a $\gamma_{c}$-null set $N=\limsup_{i}A_{i}$. For $x\notin N$ there is $i_{0}$ with $x\notin A_{i}$ for $i\ge i_{0}$, that is $|u_{j_{i}}(x)-f(x)|<2^{-i}$, so $u_{j_{i}}(x)\to f(x)$.

For $i\in\mathbb{N}$ let $w_{i}=\mathbf{1}_{X\setminus N}\,|u_{j_{i}}|^{q}$, a nonnegative Borel function by Power-Integrable Functions and the p-Seminorm §measurable-power and Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $x\notin N$, $w_{i}(x)\to|f(x)|^{q}$ by Properties of Real Powers of Nonnegative Real Numbers §continuity, so $\liminf_{i}w_{i}(x)=|f(x)|^{q}$; for $x\in N$, $\liminf_{i}w_{i}(x)=0$. Hence $\liminf_{i}w_{i}=|f|^{q}$ almost everywhere, and by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, Fatou's Lemma and Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\int_{X}|f|^{q}\,d\gamma_{c}=\int_{X}\liminf_{i}w_{i}\,d\gamma_{c}\le\liminf_{i}\int_{X}w_{i}\,d\gamma_{c}\le\liminf_{i}\lVert u_{j_{i}}\rVert_{q}^{q}\le\liminf_{i}\bigl((q-1)^{m/2}\lVert F_{j_{i}}\rVert_{2}\bigr)^{q},$$

the last step by (1) and Properties of Real Powers of Nonnegative Real Numbers §monotone. By Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions §minkowski, together with $\lVert F-F_{j}\rVert_{2}=\lVert F_{j}-F\rVert_{2}$ from Elementary Properties of the p-Seminorm §homogeneous (with $c=-1$), $|\lVert F_{j}\rVert_{2}-\lVert F\rVert_{2}|\le\lVert F_{j}-F\rVert_{2}\to0$, so by Properties of Real Powers of Nonnegative Real Numbers §continuity the right-hand side equals $((q-1)^{m/2}\lVert F\rVert_{2})^{q}$. Thus $\int_{X}|f|^{q}\,d\gamma_{c}<\infty$, so $F\in L^{q}(\gamma_{c})$, and taking powers $1/q$ (Properties of Real Powers of Nonnegative Real Numbers §monotone, Properties of Real Powers of Nonnegative Real Numbers §inverse) gives $\lVert F\rVert_{q}\le(q-1)^{m/2}\lVert F\rVert_{2}$.

Step 4 (Claim 2). Let $m\ge1$, $\sigma>0$ with $\lVert F\rVert_{2}\le\sigma$, and $s\ge(2\mathrm{e})^{m/2}\sigma$. Put $q=\mathrm{e}^{-1}(s/\sigma)^{2/m}$. By Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §exponents, $(s/\sigma)^{2/m}\ge((2\mathrm{e})^{m/2})^{2/m}=2\mathrm{e}$, so $q\ge2$. By Properties of Real Powers of Nonnegative Real Numbers §monotone, $|f(x)|\ge s$ if and only if $|f(x)|^{q}\ge s^{q}$; so The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §markov, applied to $|f|^{q}$ with $s^{q}>0$, and claim 1 give

$$s^{q}\,\gamma_{c}\bigl(\{x\in X:|f(x)|\ge s\}\bigr)\le\int_{X}|f|^{q}\,d\gamma_{c}=\lVert F\rVert_{q}^{q}\le\bigl((q-1)^{m/2}\sigma\bigr)^{q}\le\bigl(q^{m/2}\sigma\bigr)^{q},$$

using $\lVert F\rVert_{2}\le\sigma$, $q-1<q$ and Properties of Real Powers of Nonnegative Real Numbers §monotone. By Properties of Real Powers of Nonnegative Real Numbers §product and Properties of Real Powers of Nonnegative Real Numbers §exponents, $q^{m/2}=((s/\sigma)^{2/m})^{m/2}(\mathrm{e}^{-1})^{m/2}=(s/\sigma)\exp(-m/2)$, since $(\mathrm{e}^{-1})^{m/2}=\exp(\frac{m}{2}\log\exp(-1))=\exp(-m/2)$ by Real Power of a Positive Real Number and $\mathrm{e}^{-1}=\exp(-1)$. Hence $q^{m/2}\sigma=s\exp(-m/2)$, and by Properties of Real Powers of Nonnegative Real Numbers §product and Real Power of a Positive Real Number, $(q^{m/2}\sigma)^{q}=s^{q}\exp(-m/2)^{q}=s^{q}\exp(-mq/2)$. Dividing by $s^{q}>0$,

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

This proves claim 2.