There is an excellent paper by David Warren, published in 1986 in the Journal of Hydrology, "Outflow Skewness in non-seasonal linear reservoirs with gamma-distributed inflows" (Volume 85, pp127-137; http://www.sciencedirect.com/science/article/pii/0022169486900806#). In the paper, Warren develops a method to derive analytic expressions for the cumulants of the time series of outflows from a linear reservoir that has gamma-distributed (and autocorrelated) inflows. The paper derives expressions for the first, second and third cumulants, but goes no further.
I am interested in the fourth cumulant, but am unable to duplicate Warren's method to produce the existing results on p135-137, where the expressions are derived. This is the use of equation 10, in conjunction with equations 9 and 11, to produce the three expressions A1, A2 and A3.
In the paper, Warren states that the increased complexity of terms with each successive cumulant meant that he did not continue to the fourth. However, this is a key parameter of interest, so any assistance would be much appreciated!