Find the variance-covariance matrix for a linear combination of multiple bivariate normal distribution? I have an arbitrary number of independnet bivariate normal distributions with $\mu_i = [x_i,z_i]$ & $ \Sigma_i=
\left(\begin{array}{cc} 
\sigma^2_{x_i} & \sigma^2_{x_i,z_i}\\\ 
\sigma^2_{x_i, z_i} & \sigma^2_{z_i}
\end{array}\right)
$
Where i is arbitrarily large
I want to take a linear combination of these bivariate normal distributions with weights $c = [c_1,...,c_i]$ where $\sum c_i = 1$ & $c_i >0$
Obviously, the linear combination of $\mu_{mixture} = [\sum c_ix_i,\sum c_iz_i]$
However, I am not sure about the linear combination of the variance-covariance matrix.
Does anyone know how I can calculate this pooled & weighted variance-covariance? Looking for the variance-covariance matrix for the mixture distribution.
Thanks so much!

Thanks to @jld & @whuber for their answeres. Both were super helpful as I tried to solve this problem. With continued research, I found the post which I'll link below. It has the same info that jld & whuber shared, but it helped me solve it so I wanted to link it here
https://math.stackexchange.com/questions/195911/calculation-of-the-covariance-of-gaussian-mixtures
