Proving Linear Filtering of Gaussian Process Still Gaussian

[chingkui]

How to prove the output of Linear Filtering a Gaussian Process is still Gaussian? It has been stated in many books I read, but none of them actually prove it. One even stated that "The technical mechinery to prove this property is beyond the scope of this book..."
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.

 

[Inquisitive_Mind]

I suspect that the a formal proof will be quite technical. However, to understand it intuitively, one may try to discretise the integral in the definition of y(t). The decretised integral is a linear combination of jointly Gaussian variables, and is therefore Gaussian. Each of the y(ti) can be discretised this way, and each is a linear combination of a set of jointly Gaussian variables, and {y(ti)} is therefore jointly Gaussian.

Of course this is far from being a proof. But I think this theorem belongs to the category where it can be easily understood but not easily proved.

 

[tacman]

Proving that the output of linear filtering a Gaussian process is still Gaussian requires a deep understanding of mathematical concepts and techniques such as probability theory, stochastic processes, and linear algebra. It is not a trivial task and may require advanced mathematical knowledge.

To prove this statement, one would need to use the properties of Gaussian processes, such as the definition of a Gaussian process and its covariance function. Additionally, one would need to use the properties of linear filtering, such as the convolution operation and its properties.

The proof would involve showing that the output of the linear filtering operation, which is the convolution of the Gaussian process with a function h(t), is also a Gaussian process. This can be done by showing that the output satisfies the definition of a Gaussian process, which states that any finite number of points in the process should be jointly Gaussian.

To do this, one would need to use techniques such as moment generating functions, characteristic functions, and the properties of Gaussian distributions. The proof may also involve using properties of integrals and limits to show that the output of the convolution operation is still Gaussian.

Overall, proving that the output of linear filtering a Gaussian process is still Gaussian is a complex task that requires a deep understanding of mathematical concepts and techniques. It is not something that can be easily explained or proved without a strong mathematical background. Therefore, it is understandable that many books may state this property without providing a proof.