Linear Transformations & the Multivariate Generating Function

TL;DR

This paper explores how linear transformations of multi-indexed sequences affect their multivariate generating functions, with applications to distributions and moments of conditioned non-negative discrete random variables.

Contribution

It derives a formula for the multivariate generating function of linear combinations of sequences, extending analysis tools for multivariate distributions.

Findings

Derived the multivariate generating function for linear combinations.

Applied results to Poisson and multinomial distributions.

Provided methods for calculating conditioned distributions and moments.

Abstract

This note examines linear combinations of multi-indexed sequences and derives the multivariate generating function of such a linear combination in terms of the original sequence's m.g.f. Applications include finding distributions and moments of non-negative discrete random variables conditioned on non-negative linear combinations of the original variables. Examples include independent Poisson r.v.'s and a $d$-variate multinomial distribution.