In circular statistics, the expectation value of a random variable $Z$ with values on the circle $S$ is defined as $$ m_1(Z)=\int_S z P^Z(\theta)\textrm{d}\theta $$ (see wikipedia). This is a very natural definition, as is the definition of the variance $$ \mathrm{Var}(Z)=1-|m_1(Z)|. $$ So we didn't need a second moment in order to define the variance!
Nonetheless, we define the higher moments $$ m_n(Z)=\int_S z^n P^Z(\theta)\textrm{d}\theta. $$ I admit that this looks rather natural as well at first sight, and very similar to the definition in linear statistics. But still I feel a little bit uncomfortable, and have the following
Questions:
1. What is measured by the higher moments defined above (intuitively)? Which properties of the distribution can be characterized by their moments?
2. In the computation of the higher moments we use multiplication of complex numbers, although we think of the values of our random variables merely as vectors in the plane or as angles. I know that complex multiplication is essentially addition of angles in this case, but still: Why is complex multiplication a meaningful operation for circular data?