Periodic distribution functions - Finding maximum (expected?) value.

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If I had some normal gaussian distributions, with covariance matrices in N dimensions I could easily multiply them to get a resultant distribution (often used in a Kalman filters)

I have a probability distribution that is periodic, such as that linked: periodic distribution.

If many of them are combined you get something like this: multiple periodic distributions.

1) Is there a nice way to determine where the bright spot in the middle is (and covariance) without iteration over the entire search space?

2) What are the proper terms for what I've described?