On the Determinants of Moment Matrices

Abstract

An investigation is carried out in the behavior of the determinants of certain moment matrices, for which the $(i, j)$ entry is the $(i + j)$th moment of a distribution $F$. The determinant can be represented as the expected value of a $U$-statistic type kernel. The structure of the kernel illustrates how the determinant carries information about the number of support points of the distribution $F$. The kernel representation can be extended to the determinant of a matrix of moment generating function derivatives, where the $(i, j)$ entry is the $i + j$th derivative of the moment generating function of $F$. When done, this reveals that this determinant is itself, as a function of $t$, a moment generating function. When this somewhat surprising result is applied to members of the quadratic variance exponential family, one obtains the result that they are closed under this two-step operation of taking derivatives, then computing determinants. This results in an elementary recursion for the values of the moment determinants. The final result gives the convergence of the moment determinants to the normal theory values under central limit theorem conditions.