PreprintCC BY 4.0arXiv
Exact Asymptotics for the Harmonic Descent Chain and Extended Convergence Bounds for Recursive Statistics
Abstract
We give an exact asymptotic expansion of the occupation probabilities a(n,k) of the harmonic descent chain, introduced by D. Aldous, as n→∞ with k∈o(√n). In addition, we extend the recent results of Brandenberger et al. by giving a bound on the rate of convergence of sequences defined recursively by xn = 1/(Hn−1)∑i=1n−1xi/(n−i) for more general initial conditions. This result enables the application of an existing central limit theorem, also by Brandenberger et al., to statistics of the critical beta-splitting tree which satisfy our more general initial conditions.
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BibTeX
@misc{svihla2026exact,
title = {{Exact Asymptotics for the Harmonic Descent Chain and Extended Convergence Bounds for Recursive Statistics}},
author = {Sean P. Svihla},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00347},
eprint = {2611.00347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00347},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Asymptotics for the Harmonic Descent Chain and Extended Convergence Bounds for Recursive Statistics
AU - Sean P. Svihla
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00347
UR - https://arxiv.org/abs/2611.00347
AB - We give an exact asymptotic expansion of the occupation probabilities a(n,k) of the harmonic descent chain, introduced by D. Aldous, as n→∞ with k∈o(√n). In addition, we extend the recent results of Brandenberger et al. by giving a bound on the rate of convergence of sequences defined recursively by x_n = 1/(H_n−1)∑_i=1^n−1x_i/(n−i) for more general initial conditions. This result enables the application of an existing central limit theorem, also by Brandenberger et al., to statistics of the critical beta-splitting tree which satisfy our more general initial conditions.
ER -