# Product of densities of a wrapped normal distribution

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?

• The total mass usually isn't $1$. Jul 31 '12 at 13:22
• Yes, it isn't. For the classical case of a normal distribution, it is possible to derive a formula for the new mass. I wonder whether there is a similar formula for the Wrapped Normal case. Jul 31 '12 at 16:06
• one way to check is by seeing if the product of two Jacobi Theta functions is again some kind of Jacobi theta function with "reasonable" parameters---might be good to tag this question with "special-functions" to attract the attention of special function experts. Jul 31 '12 at 20:48
• If $\mu_1 = 0$ and $\mu_2 = \pi$ and $\sigma_1=\sigma_2$ then for generic values of $\sigma_1$ the product is bimodal, which I think can't happen for a wrapped normal density. Aug 1 '12 at 6:15
• Douglas' comment answers my question. Bimodality indeed can not happen in a wrapped normal distribution. Thus, the product of two wrapped normal densities is unfortunately not wrapped normal. Aug 1 '12 at 16:43