A Linear Time, and Constant Space, Algorithm to Compute the Mixed Moments of the Multivariate Normal Distributions

TL;DR

This paper introduces a linear-time, constant-space algorithm for computing mixed moments of multivariate normal distributions, with applications in statistics and combinatorics, implemented for low-dimensional cases.

Contribution

It presents a novel recurrence-based algorithm that efficiently computes mixed moments for multivariate normal distributions with any covariance matrix.

Findings

Algorithm runs in linear time and constant space.

Successfully implemented for bivariate and trivariate cases.

Recurrences for higher dimensions remain computationally challenging.

Abstract

Using recurrences gotten from the Apagodu-Zeilberger Multivariate Almkvist-Zeilberger algorithm we present a linear-time, and constant-space, algorithm to compute the general mixed moments of the k-variate general normal distribution, with any covariance matrix, for any specific k. Besides their obvious importance in statistics, these numbers are also very significant in enumerative combinatorics, since they count in how many ways, in a species with k different genders, a bunch of individuals can all get married, keeping track of the different kinds of heterosexual marriages. We completely implement our algorithm (with an accompanying Maple package, MVNM.txt) for the bivariate and trivariate cases (and hence taking care of our own 2-sex society and a putative 3-sex society), but alas, the actual recurrences for larger k took too long for us to compute. We leave them as computational challenges.