The density of a wrapped normal distribution is given by $$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=\infty }^{\infty }\text{Exp}\left[\frac{(\theta \mu 2\pi k)^2}{2\sigma^2}\right]$$ Considering two density functions $f(x),\ g(x)$ of a wrapped normal distribution with respective parameters $\mu_1,\ \mu_2$ and $\sigma_1,\ \sigma_2$, is the product $h(x)=f(x)g(x)$ a density function of a wrapped normal distribution?

This is really a followup on Suvrit's comment. There are plenty of formulas for products of theta functions, many of them found in this Iowa State report. (see particularly page 7). Whether any of them answer the OP's question is for the OP to find out (and tell us...) 

