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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 $v$.

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Steve
    Commented Aug 3, 2021 at 21:13
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    $\begingroup$ @Steve : Yes, this is another way to come to the same conclusion. $\endgroup$ Commented Aug 3, 2021 at 21:25

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