Orthogonality and probability: beyond nearest neighbor transitions

TL;DR

This paper investigates the limitations of the Karlin-McGregor method for Markov processes beyond nearest neighbor transitions and explores potential solutions to extend its applicability.

Contribution

It analyzes why the orthogonal polynomial approach fails for complex Markov processes and proposes possible methods to overcome these limitations.

Findings

Identifies key reasons for the failure of the Karlin-McGregor method beyond nearest neighbors.

Proposes potential modifications or new approaches to extend the method.

Provides initial testing of suggested solutions.

Abstract

In this article, we will explore why Karlin-McGregor method of using orthogonal polynomials in the study of Markov processes was so successful for one dimensional nearest neighbor processes, but failed beyond nearest neighbor transitions. We will proceed by suggesting and testing possible fixtures.