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?
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$\begingroup$ The total mass usually isn't $1$. $\endgroup$– Douglas ZareCommented Jul 31, 2012 at 13:22
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$\begingroup$ 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. $\endgroup$– ostap benderCommented Jul 31, 2012 at 16:06
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$\begingroup$ 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. $\endgroup$– SuvritCommented Jul 31, 2012 at 20:48
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3$\begingroup$ 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. $\endgroup$– Douglas ZareCommented Aug 1, 2012 at 6:15
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1$\begingroup$ 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. $\endgroup$– ostap benderCommented Aug 1, 2012 at 16:43
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This is really a follow-up 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...)
answered Aug 1, 2012 at 2:47