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 $v$.