Is an exponential family closed under convolution? Let 
$$
y = x_1 + x_2 + x_3
$$
where $x_1, x_2, x_3$ are draws of random variables of the same family and different parameters:  $p(x_1| \theta_1), p(x_2| \theta_2), p(x_3| \theta_3)$ 
We know that, for some distributions that are closed under convolution, then the distribution over $y$ belongs to the same distribution.
My question concerns the case where $x_i$ belong to some exponential family of distributions: does the sum of random variables from an exponential family also belong to some exponential family?
 A: The property that you describe refers to stable distributions 
You can study these convolutions of pdf's by products of characteristic functions.  For instance:


*

*The Cauchy-Lorentz distribution is a stable distribution $$E(e^{itX}) = e^{it\mu_X+\sigma_X\vert t\vert}$$ and $$E(e^{itX+Y}) = e^{it\mu_X+\sigma_X\vert t\vert} e^{it\mu_X+\sigma_X\vert t\vert} =e^{it(\mu_X+\mu_Y)+(\sigma_X+\sigma_Y)\vert t\vert} = e^{it\mu_Z+\sigma_Z\vert t\vert} $$ with $\mu_Z=\mu_X+\mu_Y$ and $\sigma_Z=\sigma_X+\sigma_y$

*but, for instance, the Gamma distribution is not a stable distribution $$E(e^{itX}) = (1-it\theta_X)^{k_X}$$ and $$E(e^{itX+Y}) = (1-it\theta_X)^{k_X} (1-it\theta_Y)^{k_Y} \neq (1-it\theta_Z)^{k_Z}  $$ where we can not make this final step (unless $\theta_X = \theta_Y$)

So for a particular distribution that is in the exponential family it is not (generally) true that is a stable distribution. 
Still you may wonder whether the sum of any two distributions in the exponential family could be any other distribution in the exponential family. 


*

*There is no general description of the characteristic function of an exponential family distribution $f(x) = h(x)g(\theta) e^{\eta(\theta) \cdot T(x)}$, and trying to work out the convolution for the general case is messy (and I don't think it works). 
Possibly one could work out a particular case to show that a convolution is yielding some distribution that is not a exponential family.

*If the sufficient statistic T(x) is a linear function then at least the cumulant generating function for T might be used to proof some stability property:
$$k(u \vert \eta) = A(\eta + u) + A(\eta)$$
by using the property $k(X+Y) = k(X) + k(y)$. However, I am not sure about the inverse step, whether the cumulant generating function $k(X) + k(y)$ is necessarily referring to a valid distribution from the exponential family. But at least this is an interesting notion.
