Let $k(\cdot,\cdot)$ be a bounded kernel and $\mathcal{H}$ its associated RKHS. Define the kernel mean embedding $\mu=\int k(\cdot,x) \, dP_X(x)$ and let $\hat{\mu}=\frac{1}{n}\sum k(x_i,\cdot)$ be its sample analogue. The error between the two $\|\mu-\hat{\mu}\|_{\mathcal{H}}=O(\frac{1}{\sqrt{n}})$. Various results show this by using Bernstein type inequalities for Hilbert spaces.
To me this feels intuitive from a Central Limit theorem point of view. Is there a Hilbert space CLT? Is it possible to show such rate from an application of CLT?
