# Computing the fourth moment of a compound Gaussian distibution

I am working with a physical application, in which time-evolution is studied by monte-carlo evolution of Gaussian states (that is, the mean and variance of these states evolve according to coupled SDEs). Taking snapshots at some fixed timesteps separated longer than the autocorrelation-time, and repeating multiple times I am thus left with a sampled set of normal distributions in (X,P)-space. With $$<\cdot>$$ notation I will denote expectation values with respect to a given stochastic Gaussian, which I know analytically, and with $$\overline{\cdot}$$ expectations with respect to the monte-carlo sampling.

The object which I'm interested in is the Binder parameter, which relates to the fourth moment of the total distribution of P (For symmetry arguments in the model, I know that $$\overline{

}=0$$

, whereas the full distribution of $$

$$

, like the one of P, should be symmetric and either one-or bimodal dependent on some parameter values of the model). I am not so much interested in X-dependence, so I'll just leave this aside.

If I were only interested in the second moment $$\text{Var} P$$, I know this can be split in $$\text{Var}_2 P=\text{Var}

$$

and $$\text{Var}_1 P=\overline{<\delta_p\delta_p>}$$, where $$\delta_p=P-

$$

is a fluctuation within a given Gaussian, projected on the P-axis>. Because I have a list of data both for $$P$$ and $$<\delta_p\delta_p>$$, computation of the total variance is straightforward because $$\text{Var} P=\text{Var}_1 P+\text{Var}_2 P$$.

Is a a similar method possible to decompose the fourth moment?

I haven't found one so far.

The way I'm handling this at the moment, is by in every Gaussian additionally sample a single random $$\delta_p$$ and thus getting a list of independent $$P=

+\delta_p$$

. This seems to work somehow, but I'm curious if there is a better way, because the need to sample the individual Gaussians (and only once per Gaussian to ensure statistical independence) seems to increase the uncertainty (for which I'm using some Jackknife resampling).