**Question**


Given a random field $X(t)$ where the parameter space $T\subset\mathbb{R}\_N$. Is there result regarding the concentration of the random field? For example


$\mathbb{P}\{\|X(t)-\mathbb{E}\{X(t)\}\|\geq t\}\leq \text{small}$ where $\|\cdot\|$ is certain norm, maybe $L\_p$ norm.


**Take the following random function as an example**


$f(x):=f(x\_1,x\_2,x\_3)=a\_1\sin(x\_1-x\_2)+a\_2\sin(x\_2-x\_3)$, where $a\_1=1+w\_1,a\_2=1+w\_2)$ and $w\_1,w\_2$ are independent standard Gaussian scalar. The coefficients $a\_1,a\_2$ are independent, and $x\_1,x\_2\in\mathbb{R}$.


The expectation of the random polynomial is $\mathbb{E}f(x)=\sin(x\_1-x\_2)+\sin(x\_2-x\_3)$, which is not random.


Then my question is the concentration of random function to its expectation $$\mathbb{P}\{\|a\_1\sin(x\_1-x\_2)+a\_2\sin(x\_2-x\_3)-\big(\sin(x\_1-x\_2)+\sin(x\_2-x\_3)\big)\|\_{L\_p}\geq t\}\leq \text{small}$$


**What I tried**


I read the book *random fields and geometry - Robert J. Adler, Jonathan E. Taylor*, and found it focus on (at least in the chapter 2) the exeedence probability


$$\mathbb{P}\{\sup\_{t\in T}f(t)\geq u\}$$


I did not find concentration properties of random field in this book and other materials on random fields. I am not sure whether it is because I did not recognize them or there actually have no result regarding this type.


Anyone could help? Thanks!



concentration of random field to its expectation function
$\newcommand\R{\mathbb R}$For real $x\_1,x\_2,x\_3$, let  

\begin{equation}
 g(x\_1,x\_2,x\_3):=\sin(x\_1-x\_2),\quad h(x\_1,x\_2,x\_3):=\sin(x\_2-x\_3). 
\end{equation}
The request is to bound
\begin{equation}
 P(Y\ge t)
\end{equation}
from above, where
\begin{equation}
 Y:=\|w\_1 g+w\_2 h\|\_{L^p}
\end{equation}
and $w\_1,w\_2$ are independent standard normal random variables.


If $\|\cdot\|\_{L^p}$ is understood here as the standard norm on $L^p(\R^3)$, then $Y=\infty$ with probability $1$. This follows because, say, (i) $h(x\_1,x\_2,x\_3)$ is nonzero, continuous, and periodic in $x\_3$; (ii) the function $|\cdot|^p$ is continuous and has nonzero variation on any interval of nonzero length; and (iii) $P(w\_2\ne0)=1$.


So, for the problem to make sense, assume that the norm $\|\cdot\|\_{L^p}$ is the standard norm on $L^p(T)$, where $T$ is a measurable subset of $\R^3$ such that $G:=\|g\|\_{L^p(T)}\in(0,\infty)$ and $H:=\|h\|\_{L^p(T)}\in(0,\infty)$.


Then, by Minkowski's inequality, for $p\in[1,\infty]$,
\begin{equation}
\begin{aligned}
 P(Y\ge t)&\le P(G|w\_1|+H|w\_2|\ge t) \\ 
 &\le P(G|w\_1|\ge\tfrac G{G+H}\,t)+P(H|w\_2|\ge\tfrac H{G+H}\,t) \\
 &=2P(|w\_1|\ge\tfrac t{G+H}) \\ 
 &\le2\exp(-\tfrac{t^2}{2(G+H)^2}) 
\end{aligned}
\end{equation}
for real $t\ge0$.




---


It was assumed above that both $G$ and $H$ are $>0$.
If exactly one of the norms $G$ and $H$ is $0$ (and still $G<\infty$ and $H<\infty$), then, similarly, $P(Y\ge t)\le\exp(-\tfrac{t^2}{2(G+H)^2})$ for real $t\ge0$. If $G=H=0$, then, clearly, $P(Y\ge t)=0$ for real $t>0$. So, when the condition that $G>0$ and $H>0$ does not hold, we have better upper bounds than $2\exp(-\tfrac{t^2}{2(G+H)^2})$.




---


Suppose now that
\begin{equation}
 Y:=\|w\_1 g\_1+\cdots+w\_n g\_n\|\_{L^p},
\end{equation}
where $w\_1,\dots,w\_n$ are independent standard normal random variables and the $g\_j$'s are deterministic functions in $L^p$ with $G:=\max\_j\|g\_j\|\_{L^p}\in(0,\infty)$. Then for $p\in[2,\infty)$, by [Theorem 3.3](https://projecteuclid.org/journals/annals-of-probability/volume-22/issue-4/Optimum-Bounds-for-the-Distributions-of-Martingales-in-Banach-Spaces/10.1214/aop/1176988477.full) (with $\Gamma=1$; see also [typo correction](https://projecteuclid.org/journals/annals-of-probability/volume-27/issue-4/Correction--Optimum-bounds-for-the-distributions-of-martingales-in/10.1214/aop/1022874833.full)) and [Proposition 2.1](https://projecteuclid.org/journals/annals-of-probability/volume-22/issue-4/Optimum-Bounds-for-the-Distributions-of-Martingales-in-Banach-Spaces/10.1214/aop/1176988477.full), we have the following Bernstein-type bound:
\begin{equation}
 P(Y\ge t)\le2\exp\Big(-\frac{t^2}{B^2+t+B\sqrt{B^2+2t}}\Big)
\end{equation}
for real $t\ge0$, where $B:=\sqrt{(p-1)G^2 n}$. In particular, if $t=O(\sqrt n)$ or, more generally, $t=o(n)$ as $n\to\infty$, then
\begin{equation}
 P(Y\ge t)\le2\exp\Big(-\frac{t^2}{(2+o(1))B^2}\Big)
 =2\exp\Big(-\frac{t^2}{(2+o(1))(p-1)G^2 n}\Big),
\end{equation}
so that we have a normal-type upper bound on $P(Y\ge t)$ with an asymptotically correct constant factor $2+o(1)$ in the denominator of exponent (consider the special case when the $g\_j$ is the constant $1$ on (say) the cube $[0,1]^d$, so that one may take here $p=2$).


