The $x_i$ are an arbitrary sequence of points in $\cal X$, it would be great to prove upper bounds in the worst case. And $\cal H_k$ is the RKHS of kernel $k$. You can find an introduction of the link between RKHS and GP in the chapter 6 of the Rasmussen and Williams' book (available online).

Oops, I edited the question, I wanted to get high probabilistic results such as $P[\lVert \mu_n \rVert_{\cal H_k} < ...] > 1-\delta$.

We can remark that if we drop the term $\sigma^2 \mathrm{I}$, the transformation can be viewed as a projection where we remove the space generated by the $\{x_t\}_{t\leq T}$. Then, the new norm can be related to the other using Pythagorean theorem.