Convolution of two Gaussians is a Gaussian

probability probability-distributions normal-distribution

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.


  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

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


See this for two common alternatives.


I think this pdf file can help you.

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