Cumulants and partition lattices II: generalised k-statistics

The role played by the Möbius function of the lattice of all partitions of a set in the theory of k -statistics and their generalisations is pointer out and the main results conscerning these statistics are drived. The definitions and formulae for the expansion of products of generalished k -statistics are presented from this viewpoint and applied to arrays of random variables whos moments satisfy stitable symmentry constraints. Applications of the theory are given including the calculation of (joint) cumulants of k -statistics, the minimum variace estimation of (generalised) moments and the asymptotic behaviour of generalised k -statistics viewed as (reversed) martingales.