I consider the case of *independent* Gaussian (or any light-tailed, for that matter) coefficients with variances decaying like $x^{-2k}$ - note that in this case the coefficients themselves decay essentially like $x^{-k}$, up to a logarithmic correction. For this random series the answer will be $C^{k-n/2}$, again up to a logarithmic correction.

To prove this first note that the covariance function of this (stationary) process is in $C^{2k-n-\varepsilon}$ because Fourier series (of the covariance) decays like $x^{-2k}$. Then use the multidimensional version of Kolmogorov's continuity criterion, as formulated in, say, Lemma 2.1 of [Scheutzow][1], (recall that for the Gaussians all $L^p$ norms are equivalent to the $L^2$ norm and use high moments there). It will follow that a Gaussian process with $C^{\alpha}$ covariance has $C^{\alpha/2-\varepsilon}$ sample paths, hence the result.


  [1]: http://arxiv.org/abs/0804.1263