The Harmonic Descent Chain

The decreasing Markov chain on {1,2,3, \ldots} with transition probabilities $p(j,j-i)\propto 1\mathrm{/}i$ 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.