I was hoping someone knows whether the following statement is true or not.
Suppose $F$ is a given normal distribution and $G$ is a distribution that has a given mean $\mu$ and variance $\sigma^2$ (but can otherwise be anything). Then, the distribution G that has the minimum KL distance from F is a normal distribution (of course, $G$ would be a normal distribution mean $\mu$ and variance $\sigma^2$).
Is this true and if so, is there a reference? Really appreciate any assistance.