Asymptotics for the harmonic descent chain and applications to critical beta-splitting trees
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| Format: | Preprint |
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2025
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| _version_ | 1866915375887679488 |
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| author | Brandenberger, Anna Chin, Byron Mossel, Elchanan |
| author_facet | Brandenberger, Anna Chin, Byron Mossel, Elchanan |
| contents | Motivated by the connection to a probabilistic model of phylogenetic trees introduced by Aldous, we study the recursive sequence governed by the rule $x_n = \sum_{i=1}^{n-1} \frac{1}{h_{n-1}(n-i)} x_i$ where $h_{n-1} = \sum_{j=1}^{n-1} 1/j$, known as the harmonic descent chain. While it is known that this sequence converges to an explicit limit $x$, not much is known about the rate of convergence. We first show that a class of recursive sequences including the above are decreasing and use this to bound the rate of convergence. Moreover, for the harmonic descent chain we prove the asymptotic $x_n - x = n^{-γ_* + o(1)}$ for an implicit exponent $γ_*$. As a consequence, we deduce central limit theorems for various statistics of the critical beta-splitting random tree. This answers a number of questions of Aldous, Janson, and Pittel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics for the harmonic descent chain and applications to critical beta-splitting trees Brandenberger, Anna Chin, Byron Mossel, Elchanan Probability Combinatorics 60C05, 60F05, 05C05 Motivated by the connection to a probabilistic model of phylogenetic trees introduced by Aldous, we study the recursive sequence governed by the rule $x_n = \sum_{i=1}^{n-1} \frac{1}{h_{n-1}(n-i)} x_i$ where $h_{n-1} = \sum_{j=1}^{n-1} 1/j$, known as the harmonic descent chain. While it is known that this sequence converges to an explicit limit $x$, not much is known about the rate of convergence. We first show that a class of recursive sequences including the above are decreasing and use this to bound the rate of convergence. Moreover, for the harmonic descent chain we prove the asymptotic $x_n - x = n^{-γ_* + o(1)}$ for an implicit exponent $γ_*$. As a consequence, we deduce central limit theorems for various statistics of the critical beta-splitting random tree. This answers a number of questions of Aldous, Janson, and Pittel. |
| title | Asymptotics for the harmonic descent chain and applications to critical beta-splitting trees |
| topic | Probability Combinatorics 60C05, 60F05, 05C05 |
| url | https://arxiv.org/abs/2505.24821 |