# Tail bound on the RKHS norm of a zero-mean Gaussian process

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 $$v$$.

In fact, if the RKHS $$\mathcal{H}$$ is infinite dimensional, then $$\mathbb P(f\in\mathcal{H})=0$$ -- see e.g. Corollary 4.10. So, no inequality of the desired form exists in infinite dimensions.
• Thank you so much for your pointer. From Section 4.1.3 of this paper, Gaussian process can be represented as an infinite sequence $\sum_{i\in I} z_i \cdot \lambda_i ^{1/2} \phi_i$ where $z_i \sim N(0,1)$. Indeed the RKHS norm of $f\sim \mathcal{GP}$ is infinity. Commented Aug 3, 2021 at 21:13