Lancaster Interactions Revisited

Abstract

Additive interactions of $n$-dimensional random vectors $X$, as defined by Lancaster, do not necessarily vanish for $n \geq 4$ if $X$ consists of two mutually independent subvectors. This defect is corrected and an explicit formula is derived which coincides with Lancaster's definition for $n < 4$. The new definition leads also to a corrected Bahadur expansion and has certain connections to cumulants. The main technical tool is a characterization theorem for the Moebius function on arbitrary finite lattices.