The Harmonic Descent Chain

TL;DR

This paper provides a straightforward probabilistic analysis of the harmonic descent chain, a decreasing Markov chain related to beta-splitting trees, offering an accessible alternative to complex existing methods.

Contribution

It introduces a direct, nearly self-contained probabilistic approach to analyze the harmonic descent chain, simplifying the understanding of its occupation probabilities.

Findings

Derived explicit occupation probabilities for the harmonic descent chain

Provided an accessible alternative to Mellin transform-based methods

Enhanced understanding of the chain's structure and behavior

Abstract

The decreasing Markov chain on \{1,2,3, \ldots\} with transition probabilities $p(j,j-i) \propto 1/i$ arises as a key component of the analysis of the beta-splitting random tree model. We give a direct and almost self-contained "probability" treatment of its occupation probabilities, as a counterpart to a more sophisticated but perhaps opaque derivation using a limit continuum tree structure and Mellin transforms.