Try analyzing modulated communications signals including, QAM-16 and similar modulated signals. It is often that the fourth order moments of those signals provide harmonics related to their intrinsic modulating parameters. 4th order results can also be used as a feature extraction tool. So my answer is, 4th order moment is sometimes more than very helpful !..
As said before, the fourth moment is the kurtosis. The kurtosis is the parameter defining the fatness of the tails of the distribution. Kurtosis for the normal distribution is 3, in which case values beyond three standard deviations away from the mean are infrequent. If kurtosis is substantially larger than 3, then such values would be rather frequent. Hence, there is a direct relationship between kurtosis and whether a value ought to be regarded as extreme.
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