Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is the covariance kernel. A random function sampled from such a GP can also be regarded as a member of the RKHS $\mathcal{H}$ with kernel $K$. Thus, we can consider the random variable $\|f \|\_{ \mathcal{H}}$.


It would be interesting to see the tail behavior of such a random variable. That is, can we develop an inequality of the form
\begin{align}
\mathbb{P} \big ( \| f \|\_{ \mathcal{H}} > q(\delta) \big ) \leq \delta, \qquad \forall \delta \in (0,1). 
\end{align}
It would be great if we could characterize $q(\delta)$.


The motivation of this problem is from extending finite-dimensional Gaussian random vectors to infinite dimensions. For a finite-dimensional Gaussian random vector $v \sim N(0, \Sigma)$, we can easily get a tail bound for $\| v\|\_2$, the Euclidean norm of $v$.



Tail bound on the RKHS norm of a zero-mean Gaussian process
In fact, if the RKHS $\mathcal{H}$ is infinite dimensional, then $\mathbb P(f\in\mathcal{H})=0$ -- see e.g. [Corollary 4.10](https://arxiv.org/abs/1807.02582). So, no inequality of the desired form exists in infinite dimensions.


