Sum of Gaussian mixture and Gaussian scale mixture

Question

What will be the distribution of the sum of two independent random variables, say $X$ and $Y$, when $X$ has a Gaussian mixture distribution (when we take Gaussian distribution with different location parameters), and $Y$ is a Gaussian scale mixture (when we take Gaussian distributions with different scale parameters)?

Let $Z=X+Y$ then, the mixture distribution of $X$ has density $\sum_{i=1}^n \pi_i \mathcal{N}(x;\mu_i,\sigma^2_X)$ and the mixture distribution of $Y$ has density $\sum_{j=1}^m \phi_j \mathcal{N}(x;\mu_Y,\sigma^2_j)$, the symbols $\mathcal{N}(x, \dots)$ being normal densities. (One could as well let both parameters vary in both cases, $X$ and $Y$).

Answer 1

Let $X$ has the mixture distribution with density $f(x)=\sum \pi_i f_i(x)$ and $Y$ the mixture distribition with density $g(x) = \sum \phi_i g_i(x)$, and suppose $X$ and $Y$ are independent. A useful tool for analyzing distribution of sums of independent random variables is the moment generating function (look it upon wikipedia if you didn't see it yet). That is given by $\DeclareMathOperator{\E}{E} M_X(t) = \E e^{tX}$. Then the moment generating function of $X+Y$ is the product $M_X(t) M_Y(t)$. Let the moment generating functions for $X$ mixture component $i$ be $M_i$, for $Y$ mixture component $j$ be $G_j$. Then, by linearity of the expectation operator, we have $$ M_X(t) = \E e^{tX} = \int e^{tx} f(x)\; dx \\ = \int e^{tx} \sum_i \pi_i f_i(x)\; dx \\ = \sum_i \pi_i M_i(t) $$ and likewise for $Y$ $G_Y(t) = \sum_j G_j(t)$ and then the moment generating function for $X+Y$ is $$ M_X(t)G_Y(t)=\sum_i \pi_i M_i(t) \cdot \sum_j \phi_j G_j(t) \\ = \sum_i \sum_j \pi_i \phi_j M_i(t) G_j(t) $$ so the distribution of the sum is a new mixture distribution with $nm$ component (where $X$ has $n$ components, $Y$ has $m$ components), the weights are the products of the old weights $\pi_i \phi_j$, and the distribution of the component $(ij)$ is the distribution of the sum of independent random variables $X_i+Y_j$.

In your case all of those will be normal distributions, but I leave for you to work out the details. Observe that only the very last step of the argument (that left for you) pends on thse being a normal mixture. So what we have shown is a quite gneral result about the istribution of sums of random variables, each one with a mixture distribution.

Answer 2

If I understand your notation you are asking about the distribution of $Y$ when $Y = X + Z$ where $X \sim \mathcal N(a,b)$, $Z \sim \mathcal N(c,d)$.

In this case $Y$ is Gaussian if $Z$ and $X$ are independent or jointly normal, with mean $a+b$ and a covariance matrix dependent on the relationship between $X$ and $Z$.

Using the word mixture to describe the above arrangement will lead to confusion.