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I am looking for a proof for convolution of two multivariate Gaussians (where each Gaussian has multi-dimensional mean and co-variance). I found a proof in here: http://www.tina-vision.net/docs/memos/2003-003.pdf where it provides a proof for convolution of two uni-variate Gaussians and also it provides the mean and variance of the convolved PDF. Now I am looking forward to find a proof for (or extend it to) multivariate Gaussian PDFs.

Does anybody know any solution already available, or can help me with it?

Thank you.

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  • $\begingroup$ I need to add that I am looking for the mean and co-variance of the convolved PDF. So, maybe knowing the whole proof is not necessary for me. $\endgroup$
    – PickleRick
    Oct 9, 2015 at 8:49

1 Answer 1

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Perhaps the easiest way to understand convolution, in the context of probability distributions, is in terms of the sum of independent random variables. Suppose that independent random variables $X_1$ and $X_2$ have distributions $d_1$ and $d_2$ respectively. Then $X_1+X_2$ has distribution given by the convolution of $d_1$ and $d_2$. For more details on this, see for example these notes (they only deal with the univariate case but the same concepts apply equally in multivariate situations.)

In the case of two multivariate Gaussians, it is well known (e.g. by considering characteristic functions) that the sum of independent $X\sim\mathcal N(\mu_X,\Sigma_X)$ and $Y\sim\mathcal N(\mu_Y,\Sigma_Y)$ is just $X+Y\sim\mathcal N(\mu_X+\mu_Y,\Sigma_X+\Sigma_Y)$. So all you need to do is add the mean vectors and covariances matrices.

An alternative (more direct but less illuminating) approach can be found in this document.

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