Given two multivariate gaussian (say in 2D with mean $\mu$ as a 2D point and convariance marix $\Sigma$ as $2$x$2$ Matrix) $N_1(\mu_1,\Sigma_1)$ and $N_2(\mu_2,\Sigma_1)$, I would like to derive the pdf of $N_1+N_2$.
Can any one point me to the reference where i can find the pdf derivation of $N_1 + N_2$.
Thanks in advance
Best Answer
Method 1: characteristic functions
Referring to (say) the Wikipedia article on the multivariate normal distribution and using the 1D technique to compute sums in the article on sums of normal distributions, we find the log of its characteristic function is
$$i t \mu - t' \Sigma t.$$
The cf of a sum is the product of the cfs, so the logarithms add. This tells us the cf of the sum of two independent MVN distributions (indexed by 1 and 2) has a logarithm equal to
$$i t (\mu_1 + \mu_2) - t' (\Sigma_1 + \Sigma_2) t.$$
Because the cf uniquely determines the distribution we can immediately read off that the sum is MVN with mean $\mu_1 + \mu_2$ and variance $\Sigma_1 + \Sigma_2$.
Method 2: Linear combinations
View the pair of MVN distributions as being a single MVN with mean $(\mu_1, \mu_2)$ and covariance $\Sigma_1 \oplus \Sigma_2$. In block matrix form this is
$$\Sigma_1 \oplus \Sigma_2 = \pmatrix{\Sigma_1 & 0 \\ 0 & \Sigma_2}$$
where the zeros represent square matrices of zeros (indicating all covariances between any component of distribution 1 and any component of distribution 2 are zero).
The sum is given by a linear transformation and therefore is MVN. The covariance again works out to $\Sigma_1 + \Sigma_2$. (See p. 2 #4 in course notes by the late Dr. E.B. Moser, LSU EXST 7037. Edit Jan 2017: alas, the university appears to have removed them from its Web site. A copy of the original PDF file is available on archive.org.)