Let $X_t$ a continuous Gaussian process (on $[0,1]$ if it helps) with explicit covariance function. I would like to get some information about the tail probability: $$ \mathbb{P}( ||X_t||^2_{L^2} > u).$$
Is there an existing method ou result ? How to tackle this problem ?
For $s$ and $t$ in $T:=[0,1]$, let $d(s,t):=\sqrt{E(X_s-X_t)^2}$. For $t\in T$ and $S\subseteq T$, let $d(t,S):=\inf_{s\in S}d(t,s)$. Let $N_0:=1$ and $N_n:=2^{2^n}$ for natural $n$. Let $$e_n:=\inf\,\sup_{t\in T}d(t,T_n),$$ where the infimum is taken over all subsets $T_n$ of $T$ of cardinality $\le N_n$.
Then, assuming that $EX_t=0$ for all $t\in T$, for some universal positive real constant $C$ and all real $u>0$ we have
$$P\Big(\sup_{t\in T}|X_t|\ge Cu\sum_{n\ge0}2^{n/2}e_n\Big)\le2\exp(-u^2).$$
where $C$ is a universal positive real constant; cf. inequality (2.47) and Corollary 2.3.2 of Talagrand's book, so that
$$P\Big(\|X\|_2\ge Cu\sum_{n\ge0}2^{n/2}e_n\Big)\le2\exp(-u^2).$$
Another approach to this problem is to realize that, for any finite $T_n\subset T$ and any positive "weight" function $w_n$ on $T_n$, $G_n:=\|X\|_{T_n,2}^2:=\sum_{t\in T_n}w_n(t)X_t^2$ is a positive-semidefinite quadratic form in centered Gaussian random variables, and then use the upper and lower bounds on the tail probabilities of $G_n$ obtained by Székely and Bakirov. Note also here that we can approximate $\|X\|_2^2$ by $\|X\|_{T_n,2}^2$.
