# Moments, Cumulants, and effective action?

1. Apr 2, 2012

### TriTertButoxy

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Hi,
I'm studying quantum mechanics and statistical mechanics, and they make heavy use of the 'correlation functions/green's functions' which are merely the moments of the distribution of some variable.

I have very intuitive understanding of moments and cumulants in terms of the distribution. The moment of a distribution is the sum (or integral) over the distribution weighted by the power of that variable, and the cumulants carry the 'essence' of the shape of the distribution (mean, variance, skewness, ...). In fact, it is very useful to define generating functions of moments Z(j) and cumulants W(j). But then, there's this new generating function (the effective action), defined as the Legendre transform of the generating function of cumulants, W(j):

$$\Gamma(x)=xj-W(j)=\sum_{n=0}^\infty\gamma_n \frac{1}{n!}x^n$$

The diagrammatic interpretation of this is clear -- they generate 1PI graphs. But what do they mean in statistics? What do they characterize about the distribution?
Or more specifically, what do the individual $\gamma_n$ characterize about the distribution?

Thanks! and even more thanks!