I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the convolution of two Gaussians is a Gaussian.
- the Fourier transform (FT) of a Gaussian is also a Gaussian
- The convolution in frequency domain (FT domain) transforms into a simple product
- then taking the FT of 2 Gaussians individually, then making the product you get a (scaled) Gaussian and finally taking the inverse FT you get the Gaussian
See this for two common alternatives.
I think this pdf file can help you.
There is a proof for product of multivariate Gaussian PDFs in here. Maybe this can help: http://www.tina-vision.net/docs/memos/2003-003.pdf