Asymptotic normality for general subtree counts in conditioned Galton--Watson trees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911498792599552 |
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| author | Rakotoniaina, Fameno Ralaivaosaona, Dimbinaina |
| author_facet | Rakotoniaina, Fameno Ralaivaosaona, Dimbinaina |
| contents | Let $\mathcal{T}$ denote a Galton--Watson tree with offspring distribution $ξ$ satisfying $\mathbb{E}(ξ) = 1$, and let $\mathcal{T}_n$ be the Galton--Watson tree conditioned to have exactly $n$ nodes. We show that, under a mild moment condition on $ξ$, the number of occurrences of a fixed rooted plane tree $\mathbf{t}$ as a general subtree in $\mathcal{T}_n$ is asymptotically normal as $n \to \infty$, with both mean and variance linear in $n$. In addition, we prove that this limiting distribution is nondegenerate except for some special cases where the variance remains bounded. These results confirm a conjecture of Janson in recent work on the same topic. Finally, we present examples showing that if the proposed moment condition on $ξ$ is violated, the conclusion may fail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08076 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic normality for general subtree counts in conditioned Galton--Watson trees Rakotoniaina, Fameno Ralaivaosaona, Dimbinaina Probability Combinatorics Primary 60C05, 60J80, Secondary 05C05, 05C80 Let $\mathcal{T}$ denote a Galton--Watson tree with offspring distribution $ξ$ satisfying $\mathbb{E}(ξ) = 1$, and let $\mathcal{T}_n$ be the Galton--Watson tree conditioned to have exactly $n$ nodes. We show that, under a mild moment condition on $ξ$, the number of occurrences of a fixed rooted plane tree $\mathbf{t}$ as a general subtree in $\mathcal{T}_n$ is asymptotically normal as $n \to \infty$, with both mean and variance linear in $n$. In addition, we prove that this limiting distribution is nondegenerate except for some special cases where the variance remains bounded. These results confirm a conjecture of Janson in recent work on the same topic. Finally, we present examples showing that if the proposed moment condition on $ξ$ is violated, the conclusion may fail. |
| title | Asymptotic normality for general subtree counts in conditioned Galton--Watson trees |
| topic | Probability Combinatorics Primary 60C05, 60J80, Secondary 05C05, 05C80 |
| url | https://arxiv.org/abs/2603.08076 |