and does not level off. Only for the case of 6 moments does the variance reach a stable value, and even then the value is about three times higher than that predicted by the SSA simulations. Figure \[p53figure2\]c shows the comparison of the skewness of the SSA distribution to the skewness of a normal distribution ($\gamma=0$). For three time points where the skewness is relatively large (indicated by d, e, f) we display the histogram of the 100,000 SSA realisations together with the probability density of the normal distribution (cyan line) calculated based on the mean and variance of the SSA and the 6 moment approximation. Additionally, we plot the skew-normal distribution calculated using the mean, variance and third central moment; this is defined by the probability density function $$\begin{aligned}
f(x)=\frac{2}{\omega}\phi\left(\frac{x-\xi}{\omega}\right)\psi\left(\alpha\left(\frac{x-\xi}{\omega}\right)\right), \end{aligned}$$ with $\xi$ a location parameter and $\omega$ a scale parameter. The parameter $\delta$ can be calculated from the estimated skewness using the relation $$\begin{aligned}
|\delta|=\sqrt{\frac{\pi}{2}\frac{|\hat{\gamma}|^{2/3}}{|\hat{\gamma}|^{2/3}+((4-\pi)/2)^{2/3}}}.\end{aligned}$$ The parameter $\alpha$ can be calculated from $\delta$ with $\delta=\alpha/(\sqrt{1-\delta^2})$, and $\omega$ can be calculated from the variance using $\sigma^2=\omega^2(1-2\delta^2/\pi)$. These plots confirm what we saw above, that using only the mean and variance does not capture the full distribution in this case, 