Perhaps a related question is [Why are matrices ubiquitous but hypermatrices rare?][1] in the sense that higher-order (mixed) moments can be regarded as hypermatrices or tensors. For example, the second order moment of a random variable $x\in\mathbb{R}^n$ might be written $\mathbb{E}[x\otimes x]$. The third order moment is $\mathbb{E}[x\otimes x\otimes x]$, and so on.

The (relative) lack of extensive use of higher order moments may be because questions about them inherit the difficulties that come with all tensor problems ([Most tensor problems are NP-hard][2]).

A very clear example of higher order moments in use is independent component analysis (ICA), which essentially amounts to a tensor decomposition of the fourth order cumulant (see [here][3], for example). ICA is an instance of a slightly more general problem known as blind source separation, which has wide-ranging applications.


  [1]: https://mathoverflow.net/questions/48045/why-are-matrices-ubiquitous-but-hypermatrices-rare
  [2]: http://arxiv.org/abs/0911.1393
  [3]: ftp://ftp.esat.kuleuven.be/pub/SISTA/delathauwer/reports/ldl-99-26.pdf