Convolution of two Gaussians is a Gaussian

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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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5 Answers

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

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See this for two common alternatives.

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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:

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