We can define Student process exactly at the same as Gaussian process see here for exemple, we write $f\sim \mathcal{TP}(\nu,\Phi,k).$
Now in the same paper there is lemm2:
Suppose that $f\sim \mathcal{TP}(\nu,\Phi,k)$ and $g\sim \mathcal{G}(\Phi,k)$ (gaussian process) then $f$ tends to $g$ in distribution as $\nu\to+\infty.$
The proof starts by: It is sufficient to show convergence in density for any finite collection of inputs.
I don't understand why it's true. I am not really familiar with convergence in distribution for stochastic process but as I can remember I must have finite dimensional convergence+tightness, isn't it ?
It depends on what is understood by "convergence in distribution". For random processes, by default, the convergence in finite dimensional distributions is meant; there are so many different topologies on the sample space, so you need to write it explicitly when you want to speak of a particular one.