[Submitted on 17 Mar 2026 (v1), last revised 4 Aug 2026 (this version, v3)]

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Abstract:We establish a quenched functional central limit theorem for the total number of components of random partitions induced by a Chinese restaurant process with parameters $(\alpha,\theta), \alpha\in(0,1), \theta>-\alpha$. With $P_j$ denoting the asymptotic frequency of the $j$-th table, it is well-known that the component count has the same law as the occupancy count of an infinite urn scheme with sampling frequencies being $(P_j)_{j\in\mathbb N}$. Our analysis follows this approach and is based on earlier results of Karlin (1967) and Durieu and Wang (2016). In words, our result reveals that the fluctuations of the component count consist of two parts, one due to the sampling effect given the asymptotic frequencies $(P_j)_{j\in\mathbb N}$, the other due to the fluctuations of the random asymptotic frequencies, and in the limit the fluctuations of the two parts are conditionally independent given the $\alpha$-diversity. Our result strengthens a recent central limit theorem obtained by Bercu and Favaro (2024) via a different method.

Submission history

From: Yizao Wang [view email]
[v1] Tue, 17 Mar 2026 12:09:13 UTC (20 KB)
[v2] Tue, 7 Apr 2026 11:43:24 UTC (21 KB)
[v3] Tue, 4 Aug 2026 17:30:09 UTC (66 KB)

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