Marking and re-marking

TL;DR

Marking extends binomial thinning to a random total $X$ via multinomial marking, producing a multivariate count $\mathbf Y$ with coordinates that are generally dependent. The authors develop a unifying factorial moment generating function framework to derive moments, dispersions, and distributional forms for marking and re-marking, showing, for example, $\mathbb E \mathbf Y = \mathsf A \mathbb E \mathbf X$ and $\Phi_{\mathbf Y}(\mathbf t) = \Phi_{\mathbf X}(\mathsf A^\top \mathbf t)$. They show independence across coordinates only in the Poisson case and demonstrate that multivariate re-marking is the discrete analogue of linear transformation, with $\mathsf A \circ \mathbf X$ transforming Poisson and Hermite families accordingly. The work provides a rigorous toolbox for analyzing discrete counts with random totals under complex marking schemes, with potential applications to marked point processes and discrete transformations.

Abstract

A random number of items each independently marked with one of a collection of colours gives rise to the multinomial marking, which generalises binomial thinning. A multivariate version, where previously marked items are then re-marked, has similar properties to taking a linear transformation of a random vector.

Theorems & Definitions (16)

• Definition 1
• Example 2
• Example 3
• Example 4
• Example 5
• Theorem 6
• proof : Proof of Theorem \ref{['th:prop']}
• Example 7
• Theorem 8
• proof