pr.probability – Convolution of two Gaussian mixture model


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 $Z$?