When does a Gaussian Process not suited to my data? I am interested in fitting Gaussian Processes and multi-output Gaussian Processes on my time series. Which criteria should I check before doing so to be sure that these kind of processes are indeed relevant for the specific kind of data I am studying?
I hope there are better ways than checking that any finite linear combination of samples has a joint Gaussian distribution?
 A: Generally speaking, while using Gaussian Processes (GPs), the first thing practitioners are concerned about is the dimension of the parameter space. This is because training a GP regression model involves matrix inversions and can be computationally expensive. So if you have lots of high dimensional data to train on, it may be quite expensive. 
As for the fitting of the data, a GP will provide you with the posterior predictive distribution for the whole function space. It does not mean that it would be a good approximation of the underlying function. This largely depends on the chosen kernel (a.k.a co-variance function), and the performance of which may be inspected in terms of Mean Squared Error (MSE). I would refer to the this thesis for more details on choosing a kernel that may represent your problem. Also, please refer to this book for an extensive introduction to GPs.
On a related note, multi-output GPs might not be a good idea. As GP (hyper-)parameters are unknown, multi-variate GPs would require more parameters to be estimated. Therefore, sometimes the uncertainty estimation tend to be large. There is a paper by Kleijnen and Mehdad, which presents some evidence towards this argument (due to posting restrictions I could not add a link to it). With this, you may be better off using independent GP models for each of your functions. 
I understand, this answer may not address your question in its entirety, but I hope it provides some helpful clues. 
