Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
Hint: what's the Fourier transform of the convolution of two functions? –  Zarrax Jan 23 '11 at 20:33
The first assertion is not true. –  Shai Covo Jan 23 '11 at 20:49
@Shai: Yes it is –  Amit Jan 23 '11 at 21:28
@Amit: No (at least, for independent random variables). Why do you think it is? –  Shai Covo Jan 23 '11 at 21:32
As @sivaram suggested, taking the FT of both Gaussians, multiplying them, and IFTing the product yields the convolution of both Gaussians, which is a Gaussian in itself. That means that the FT of any Gaussian is a Gaussian (true), and that the product of both FTs (which are both Gaussians) is also a Gaussian, therefore, the product of any Gaussian is a Gaussian. –  Amit Jan 23 '11 at 22:04

4 Answers 4

up vote 2 down vote accepted

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

share|improve this answer
I know that the product of the two FT of Gaussians is also a gaussian, and that is also equivalent to the FT of the convolution of two gaussians. Do you think, to show that the convolution of two Gaussians is a gaussian, it would be easiest to take the FT of both, multiply, and take the IFT of the product? –  Amit Jan 23 '11 at 20:55
yes :) [ some extra characters to reach the minimum ] –  Zarrax Jan 23 '11 at 21:01
@Mait: Yes that is the best way out. –  user17762 Jan 23 '11 at 21:23
  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
share|improve this answer

I think this pdf file can help you.

share|improve this answer

See this for two common alternatives.

share|improve this answer
That link is great. Thank you. –  Amit Jan 23 '11 at 21:08

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.