> are these dummy implementations intended?

Yes.  There is no state.  The only parameter is the dimensionality
which is a template parameter.

We do often serialize underlying helper distributions, in your case the normal distribution _M_n. While the normal distribution mean and stddev are trivial for your case (not actually needing serialization)
the normal distribution has these _M_saved, etc. shat we should store.

So I would just serialize _M_n here.

This is a great distribution.
Thanks!

Are you looking at the normal distribution equivalent of these?

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