Alternate strategies to central moments to characterise distributions Assume a single real-valued variable, and frequent irregular observations of its value over a series of time-spans.  In each time span, I'm assuming the samples are from a distribution of values which the variable assumes during that time span... I intend to compare distributions between time-spans, hoping to find patterns that are otherwise obscured by the detail of the irregular sampling.
The most obvious way I can see to characterise these distributions is the first n central moments... so, for example, I can calculate approximations of the Mean, Variance, Skew and Kurtosis for the sample data.  I am aware, however, of distributions which do not have meaningful definitions of central moments (Cauchy distributions, for example) and recognise that trying to characterise such a distribution by approximated central moments is unlikely to provide useful insights.
Aside from central moments, what other approaches might I use to try to classify distributions?
 A: Characteristic functions are often used.  The characteristic function of the probability distribution of a real random variable $X$ is
$$
\psi_X(t) = \mathbb{E}\left(e^{itX}\right).
$$
If $t$ is real, this always exists.  There's an immense literature on this topic.
In one sense this next topic won't answer your question: you seem to want something that is still there when moments don't exist.  But as an alternative to central moments these are of interest.  The $n$th cumulant $\operatorname{cum}_n(X)$ of a probability distribution of a random variable exists only if the $n$th moment exists.  It shares with the $n$th moment and the $n$th central moment the property of $n$th-degree homogeneity: $\operatorname{cum}_n(cX) = c^n \operatorname{cum}_n(X)$.  When $n>1$, it shares with the $n$th central moment the property of translation-invariance: $\operatorname{cum}_n(X+c)=\operatorname{cum}_n(X)$.  (If $n=1$, we have equivariance rather than invariance: $\operatorname{cum}_1(X+c)=\operatorname{cum}_1(X)+c$.)  Finally, it shares with the second and third central moments the "cumulative property": for $X_1,\ldots,X_n$ independent, we have $\operatorname{cum}_n(X_x+\cdots+X_n)=\operatorname{cum}_n(X_1)+\cdots+\operatorname{cum}_n(X_1)$.  Higher central moments, i.e. $n\ge4$, don't have that property.
The second and third cumulants are in fact the second and third central moments; the first cumulant is the expectation; the fourth cumulant is
$$
\operatorname{cum}_4(X) = \mathbb{E}\left( (X - \mathbb{E}(X))^4\right) - 3\left(\operatorname{var}(X)\right)^2.
$$
See the Wikipedia article.
