I was wondering if there are any papers that study properties of the Gaussian mixture $\nu$ defined as $$ \nu(A) = \mathbb{E}_{\sigma \sim \mu}\Big[\mathbb{P}_\sigma(A)\Big], $$ where above $\mathbb{P}_\sigma$ is the distribution of a Gaussian with mean-zero variance $\sigma^2$. (Perhaps useful to consider the "finite" case where $\mu$ places mass $\mu_j$ on $\sigma_j$, so that we can write $ \nu(A) = \sum_{j} \mu_j~N(0, \sigma_j^2). $)
I am interested in any references that study properties of this type of Gaussian mixture, such as bounds on information-theoretic divergences between two such mixtures. Note that the key restriction, as compared to general Gaussian mixtures, is that the mean among all the components is fixed at 0.