Hi Umesh,

At first, that seems to me like a very odd thing to do. What motivates it in your case?

I say that because the cumulant expansion is a series expansion like, for example, a Taylor expansion: each successive term is typically supposed to be a smaller correction than the previous one. Of course, the third cumulant implies a radially skewed distribution, which the fourth cumulant does not (i.e. the third cumulant implies that the distribution is not symmetric about some distance). If I think about it, I can perhaps construct a situation where I expect that distribution to be symmetric, but not harmonic. For example, suppose there is a rigid lattice of atoms, with other atoms arranged interstitially half-way between lattice points. If the lattice atoms are treated as nearly fixed and are all of one type, then the interstitial atoms are equally likely to be less than half-way as more than half-way from one to the next. Thus, the third cumulant is zero by symmetry. The distribution may be very far from harmonic, though; in fact, it may have two minima that are not resolved (one slightly closer to one lattice atom, one slightly closer to the other). In that case, I'd think fitting a fourth cumulant but not a third would be justified.

The bottom line is that you should have a good physical reason, having to do with symmetries, for using the fourth without the third.

--Scott Calvin

Sarah Lawrence College