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arxiv: 2503.11390 · v2 · submitted 2025-03-14 · 🧮 math.ST · math.PR· stat.TH

On continuity of Chatterjee's rank correlation and related dependence measures

Jonathan Ansari , Sebastian Fuchs This is my paper

Pith reviewed 2026-05-23 00:27 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH
keywords Chatterjee's rank correlationxiMarkov productsweak continuitycopulasdependence measureselliptical distributionsl1-norm symmetric distributions
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The pith

Chatterjee's rank correlation xi is weakly continuous under Markov products of distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard measures of concordance remain continuous when distributions converge weakly, yet Chatterjee's xi fails this property and creates problems for statistical inference. The paper shows that xi regains weak continuity when the underlying distributions are replaced by their Markov products, which are the distributions of conditionally independent copies. Copula-based criteria and conditions on conditional weak convergence are supplied as sufficient conditions for the Markov products to converge weakly. These conditions then yield continuity statements for xi itself and for related dependence measures. The results are checked explicitly inside parametric families such as multivariate elliptical and l1-norm symmetric distributions.

Core claim

xi is weakly continuous with respect to the Markov products rather than ordinary weak convergence. Several sufficient conditions are given for weak continuity of the Markov products, including copula-based criteria and conditions that rely on conditional weak convergence. These conditions imply corresponding continuity results for xi and related dependence measures, which are then verified to hold throughout the parameter spaces of standard families such as multivariate elliptical and l1-norm symmetric distributions.

What carries the argument

The Markov product, which replaces a joint distribution by the law of a pair of conditionally independent copies given the conditioning variable.

If this is right

  • xi and related dependence measures become continuous functionals under the Markov-product operation whenever the supplied copula or conditional-convergence conditions hold.
  • Continuity of xi holds throughout the parameter spaces of multivariate elliptical distributions.
  • Continuity of xi holds throughout the parameter spaces of l1-norm symmetric distributions.
  • The same continuity statements apply to several other rank-based dependence measures once the Markov-product continuity is established.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Markov-product continuity may permit consistent estimation procedures that are robust to certain forms of dependence misspecification.
  • Similar continuity arguments could be examined for other rank correlations that currently lack weak continuity.
  • The copula criteria might be checked algorithmically for new parametric families before applying the continuity results.

Load-bearing premise

The distributions under study satisfy the paper's sufficient conditions, such as the copula-based criteria or the conditional weak convergence requirements.

What would settle it

A sequence of distributions whose Markov products converge weakly, that satisfy the stated copula or conditional-convergence criteria, yet whose xi values fail to converge to the xi of the limit.

read the original abstract

While measures of concordance -- such as Spearman's rho, Kendall's tau, and Blomqvist's beta -- are continuous with respect to weak convergence, Chatterjee's rank correlation xi recently introduced in Azadkia and Chatterjee (2021) does not share this property, causing drawbacks in statistical inference as pointed out in B\"ucher and Dette (2025). As we study in this paper, xi is instead weakly continuous with respect to conditionally independent copies -- the Markov products. To establish weak continuity of Markov products, we provide several sufficient conditions, including copula-based criteria and conditions relying on the concept of conditional weak convergence in Sweeting (1989). As a consequence, we also obtain continuity results for xi and related dependence measures and verify their continuity in the parameters of standard models such as multivariate elliptical and l1-norm symmetric distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript shows that Chatterjee's rank correlation ξ fails to be continuous under standard weak convergence but is continuous under weak convergence of Markov products (conditionally independent copies). It supplies sufficient conditions for this continuity, including copula-based criteria and conditions based on Sweeting's conditional weak convergence, derives consequences for related dependence measures, and verifies parameter continuity for the families of multivariate elliptical and ℓ1-norm symmetric distributions.

Significance. If the results hold, the work supplies a usable continuity notion for ξ that resolves the inference drawbacks identified in prior literature. The copula-based and conditional-convergence criteria are general tools that may apply beyond the examples treated. Explicit verification of the criteria for two standard parametric families is a concrete strength, as is the derivation of consequences for related measures.

minor comments (3)
  1. The definition of the Markov product and its relation to conditional independence should be stated explicitly in the introduction or §2 before the main theorems are invoked.
  2. Notation for the copula-based criteria (e.g., the precise form of the condition on the copula density or distribution function) should be introduced once and used consistently in the statements of the sufficient conditions.
  3. In the applications section, a short table or explicit list of the parameter values for which continuity is verified would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance in providing a usable continuity notion for Chatterjee's ξ under Markov products, and the recommendation for minor revision. We appreciate the identification of the copula-based and conditional-convergence criteria as general tools, as well as the value placed on the explicit verifications for elliptical and ℓ1-norm symmetric families.

Circularity Check

0 steps flagged

No circularity: continuity proofs rely on external weak-convergence criteria

full rationale

The paper establishes weak continuity of Chatterjee's xi with respect to Markov products by supplying sufficient conditions drawn from copula theory and Sweeting (1989) conditional weak convergence, then verifies these conditions hold for elliptical and l1-norm symmetric families. All load-bearing steps are deductive applications of independently stated external notions; no equation reduces to a fitted input by construction, no self-citation chain carries the central claim, and the derivation does not rename or presuppose its own target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard results from probability theory (weak convergence, copula representations, conditional convergence) without introducing new free parameters, invented entities, or ad-hoc axioms beyond those already established in the cited references.

axioms (2)
  • standard math Properties of weak convergence and Markov products of copulas as defined in prior literature
    Invoked to establish the continuity framework
  • standard math Conditional weak convergence concept from Sweeting (1989)
    Used as one of the sufficient conditions

pith-pipeline@v0.9.0 · 5667 in / 1306 out tokens · 40570 ms · 2026-05-23T00:27:27.316921+00:00 · methodology

discussion (0)

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Dependence functions based on Chatterjee's rank correlation

    math.ST 2026-05 unverdicted novelty 6.0

    Introduces dependence functions φ_{(Y,X)} and κ_{(Y,X)} that extend Chatterjee's ξ by quantifying geometric concentration of the Markov product near the diagonal.

  2. Dependence functions based on Chatterjee's rank correlation

    math.ST 2026-05 unverdicted novelty 6.0

    Introduces dependence functions φ_{(Y,X)} and κ_{(Y,X)} via Markov product analysis to extend Chatterjee's rank correlation with geometric and distributional interpretations of directed stochastic dependence.

  3. On a copula product linking Wasserstein correlations and rearranged dependence measures

    math.ST 2026-04 unverdicted novelty 6.0

    A copula product T links Wasserstein correlations and rearranged dependence measures, acting as a reflection on stochastically increasing copulas and projecting onto rearranged versions via T squared.

  4. Measures of association for approximating copulas

    math.ST 2025-05 unverdicted novelty 6.0

    Closed-form expressions are derived for association measures of approximating copulas including Chatterjee's ξ, with a proof that ξ(C_n) ≤ ξ(C) and convergence as n→∞ for TP2 copulas under checkerboard approximation.

Reference graph

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