Let $U$ be an open set in $\mathbb{R}^d$. Let $f(\mathbf{x}, \mathbf{y})$ be a function defined on $U \times U$ such that for all $\mathbf{x} \in U$, $\mathbf{y} \mapsto f(\mathbf{x}, \mathbf{y})$ is a real analytic function on $U$. Let $Q$ be a probability distribution having the density $q(\mathbf{x})$. Define
$$g(\mathbf{y}) := \int_U f(\mathbf{x}, \mathbf{y}) q(\mathbf{x}) \, \mathrm{d}\mathbf{x}.$$
My question is: Under what conditions can I say that $g(\mathbf{y})$ is real analytic on $U$? I understand that a finite sum of real analytic functions is real analytic. I think one has to be more careful with an integral. Thanks for your time. (This is not a homework question.)