Mathematics > Probability
arXiv:2408.11547 (math)
[Submitted on 21 Aug 2024 (v1), last revised 8 Jul 2026 (this version, v3)]
Abstract:We give conditions under which weak convergence of a stochastic process indexed in the class of $d$-dimensional hyperrectangles is sufficient to ensure convergence in the larger class of functions of uniformly bounded Hardy-Krause variation. When applied to the empirical process, this can further be extended to derive weak convergence of V-processes indexed in the class of kernel functions which are coordinate-wise of uniformly bounded Hardy-Krause variation. Our proofs use a generalisation of the Koksma-Hlawka inequality for linear operators, allowing us to establish our results without any continuity assumptions on the functions involved. Our theory is complemented by two separate applications: First, we establish asymptotic normality of Chatterjee's rank correlation in the fully general setting. Second, we present new limit theorems for U- and V-processes of strongly mixing data.
| Comments: | This article was previously entitled `Asymptotic Normality of Chatterjee's Rank Correlation'. Improved theory on extension of process convergence. Includes formula for the limiting variance of Chatterjee's rank correlation |
| Subjects: | Probability (math.PR); Statistics Theory (math.ST) |
| MSC classes: | 62H20, 60F05, 62G20, 62G30 |
| Cite as: | arXiv:2408.11547 [math.PR] |
| (or arXiv:2408.11547v3 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2408.11547 arXiv-issued DOI via DataCite |
Submission history
From: Marius Kroll [view email]
[v1]
Wed, 21 Aug 2024 11:55:34 UTC (94 KB)
[v2]
Fri, 16 May 2025 11:02:54 UTC (48 KB)
[v3]
Wed, 8 Jul 2026 13:44:19 UTC (85 KB)
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