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By definition, a Gaussian process is a function x such that for any finite integer k, and for any arbitary time t1, t2, ..., tk, that x(t1), x(t2), ...., x(tk) are jointly Gaussian RV.

To linear filter the process x(t) means just to convolute it with a function h(t), i.e., the output y(t)=h(t)*x(t)=integrate(h(t-s)x(s)ds)

To prove the statement is to prove that for any m>0, and any time t1,..., tm, that y(t1),...,y(tm) are jointly Gaussian.

Is it that difficult to prove? What does one need to prove it?

It is quite obvious that a Gaussian RV remains Gaussian after linear filtering it, but for a Gaussian process, I am not sure what to use to prove that. Does anyone know how? Thanks.