Let's define the random vector $(\alpha_1,\dots , \alpha_d)$ which follows a mixture of Gaussian distributions whose means are $\mu_1 ,\dots , \mu_K$, whose covariance matrices are $\Sigma_1 ,\dots , \Sigma_K$ and whose proportions in the mixture are $\tau_1 ,\dots , \tau_K$.
I am looking for the distribution of the sum of the components of this vector that is to say: $\sum_{i=1}^d \alpha_i$.
To draw a point from a Gaussian mixture distribution, an index $j$ is first chosen in $1, \dots K$ according to the proportions $\tau_1 ,\dots , \tau_K$ of the mixture. Then, the point is drawn from the Gaussian distribution whose mean and covariance matrix are $\mu_j$ and $\Sigma_j$. Then, it comes to my mind that I can separate the problem into $K$ cases. In each case, the $(\alpha_1,\dots , \alpha_d)$ are drawn according to only one Gaussian distribution and the distribution of $\sum_{i=1}^d \alpha_i$ is thus $\mathcal N(\mu_k.I, I\Sigma_KI)$, where I is the vector with only ones.
From this I conclude that the distribution of $\sum_{i=1}^d \alpha_i$ is a Gaussian mixture whose means are $\mu_1.I ,\dots , \mu_K.I$, whose covariances are $I\Sigma_1I ,\dots , I\Sigma_KI$ and whose proportions in the mixture are $\tau_1 ,\dots , \tau_K$.
However, I do not think my approach is thorough.
Here are my two questions:
- Is my conclusion true ?
- Is there a more formal way to find the solution ?
Thank you for your help !