Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components

Question

Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \Omega$. Thus $f:=(f_1,\ldots,f_k)$ can be seen as a vector-valued Gaussian process with iid components at each point in space. Let $\lambda:= \mathbb E[f_1(0)^2]$. Now, given $x \in \Omega$, define $\nu(x) := \|f(x)\| := (\sum_{j=1}^k f_j(x)^2)^{1/2}$. I'm interested in concentration inequalities for $\|\nu\|_\infty := \sup_{x \in \Omega}\nu(x)$.

In the special case where $k=1$, the Borell-TIS inequality kicks-in to give

$$ \begin{split} &\forall u\ge 0,\; \mathbb P(\|\nu\|_\infty > \mathbb E[\|\nu\|_\infty] + u) \le e^{-u^2/(2\sigma^2)},\\ &\text{ where }\sigma^2 := \sup_{x \in \Omega}\mathbb E[\nu(x)^2] = \mathbb E[\nu(0)^2] = \lambda \end{split} $$

Question. How to get concentration ienqualities for $\|\nu\|_\infty$ when $k \ge 2$ ?

Some notes on the case $k \to \infty$

Define a random field $Z_k$ on $\Omega$ by setting $Z_k(x):= \dfrac{1}{\sqrt{2\lambda k}}(\|f(x)\|^2-\lambda k) = \dfrac{1}{\sqrt{2\lambda k}}(\sum_{j=1}^kf_j(x)^2-\lambda k) $, for all $x \in \Omega$.

I don't know if there is an appropriate notion of CLT which could kick-in here, to give the limiting distribution of the random field $Z_k$.

Answer 1

Consider the real-valued centered Gaussian process $(X_{t,a}\colon(t,a)\in T\times B_k)$, where $$X_{t,a}:=\sum_{j\in[k]}a_j f_j(t),$$ $T:=\Omega$, $B_k$ is the unit ball in $\mathbb R^k$, and $[k]:=\{1,\dots,k\}$. Then $$\|\nu\|_\infty=\|X\|_\infty:=\sup\{|X_{t,a}|\colon t\in T, a\in B_k\}$$ and $$EX_{t,a}^2=\sum_{j\in[k]}a_j^2 Ef_j(t)^2\le\sup_{t\in T,j\in[k]}Ef_j(t)^2=:\sigma^2.$$ So, by the Borell--TIS inequality for the process $(X_{t,a})$, for all real $u\ge0$ $$ P(\|\nu\|_\infty>E\|\nu\|_\infty+u)=P(\|X\|_\infty>E\|X\|_\infty+u)\le e^{-u^2/(2\sigma^2)}. $$