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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.

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    @wnoise: You are quite right. However, this is a source for confusion: http://newsgroups.derkeiler.com/Archive/Comp/comp.soft-sys.matlab/2009-03/msg04203.html2011-05-01

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Fourier Transform will help you out to conclude that the convolution is also a gaussian.

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    @Mait: Yes that is the best way out.2011-01-23
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See this for two common alternatives.

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    That link is great. Thank you.2011-01-23
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  1. the Fourier transform (FT) of a Gaussian is also a Gaussian
  2. The convolution in frequency domain (FT domain) transforms into a simple product
  3. 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
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I think this pdf file can help you.

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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

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    It would be good of you give a synopsis of the proof here and then provide a link.2015-10-09