Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,

$$

f(x)=sum _{k=1}^K pi _k mathcal{N}left(x|mu _k,sigma _kright)

$$

$$

g(y)=sum _{i=1}^N lambda _i mathcal{N}left(y|mu _i,sigma _iright)

$$

Now, I would like to know the PDF of another variable $Z$, where $Z=X+Y$.

Is there anyone who can write the explicit PDF of $Z$?