Vertical Dependency in Sequences of Categorical Random Variables

TL;DR

This paper introduces a generalized theory of sequential dependency in categorical random variable sequences, demonstrating how such dependencies can be modeled while maintaining identical distribution, with exponential decay of cross-covariance and a new class of generating functions.

Contribution

It extends existing models by defining a natural form of sequential dependency and introduces a class of generating functions that produce dependent yet identically distributed categorical variables.

Findings

Cross-covariance decreases exponentially with sequence distance.

Sequences generated from the class $ ext{C}_ ext{delta}$ are identically distributed and dependent.

Graphical and example illustrations of the dependency structures are provided.

Abstract

This paper develops a more general theory of sequences of dependent categorical random variables, extending the works of Korzeniowski (2013) and Traylor (2017) that studied first-kind dependency in sequences of Bernoulli and categorical random variables, respectively. A more natural form of dependency, sequential dependency, is defined and shown to retain the property of identically distributed but dependent elements in the sequence. The cross-covariance of sequentially dependent categorical random variables is proven to decrease exponentially in the dependency coefficient $\delta$ as the distance between the variables in the sequence increases. We then generalize the notion of vertical dependency to describe the relationship between a categorical random variable in a sequence and its predecessors, and define a class of generating functions for such dependency structures. The main result of the paper is that any sequence of dependent categorical random variables generated from a function in the class $\mathscr{C}_{\delta}$ that is dependency continuous yields identically distributed but dependent random variables. Finally, a graphical interpretation is given and several examples from the generalized vertical dependency class are illustrated.