In a classical noisy regression setting, let $\big(f(x)\big)_{x\in\cal X}$ be a centered Gaussian process of covariance $k$ on a compact $\cal X$, and $\mathcal{F}_n$ be the filtration generated by some noisy measurements $\big(y_i = f(x_i)+\epsilon_i\big)_{i\leq n}$ where $\epsilon_i \overset{iid}{\sim} \mathcal{N}(0,\eta^2)$. Then, let $\mu_n(x) = \mathbb{E}[f(x) \mid \mathcal{F_n}]$ be the posterior mean of the GP given the noisy observations. We know that $\mu_n$ solves the following regression in the corresponding RKHS $\cal H_k$: $$\arg\!\min_{\hat{f} \in \cal H_k} \frac 1 2 \lVert \hat{f} \rVert_{\cal H_k}^2 + \frac 1 {2\eta^2} \sum_{i=1}^n \big(y_i - \hat{f}(x_i)\big)^2.$$ Do we have an upper bound on the distribution of $\lVert \mu_n \rVert_{\cal H_k}$ ?