Detailed convergence rates for harmonic descent occupation probabilities

Determine more detailed rates of convergence for the harmonic descent-chain occupation probabilities a(n,k) toward their limiting values a(k), including regimes not resolved by the fixed-k asymptotic and the partial bounds for growing k established in the paper.

Background

The harmonic descent chain is a continuous-time Markov chain on the positive integers with transition rates from state m to state i given by (m-i)^{-1}, with state 1 absorbing. For a starting state n and target state k, a(n,k) denotes the probability that the chain ever visits k. The limiting occupation probability a(k)=lim_{n\to\infty}a(n,k) is known explicitly. The cited Open Problem 8 asks for more detailed convergence rates of a(n,k) to a(k). The paper establishes the asymptotic rate n^{-\gamma_*+o(1)} for fixed k and gives two-sided bounds in a specified growing-k regime, so the broader rate-of-convergence problem remains only partially resolved.