On the Kolmogorov-Feller weak law of large numbers for the Fr\'echet mean on non-compact symmetric spaces

Jongmin Lee; Sungkyu Jung

REVIEW 1 major objections 4 minor 1 cited by

On the Kolmogorov-Feller weak law of large numbers for the Fr\'echet mean on non-compact symmetric spaces

T0 review · 1 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Sample Fréchet means on non-compact symmetric spaces converge in probability under the Kolmogorov-Feller tail condition, even with infinite moments.

desk verdict The main theorem is a solid new WLLN for Fréchet means on non-compact symmetric spaces; the partial converse has a broken proof, but the central result holds up. read the letter →

arxiv 2509.02074 v2 pith:WQOWOKSI submitted 2025-09-02 math.PR math.STstat.TH

Jongmin Lee , Sungkyu Jung This is my paper · ORCID

classification math.PRmath.STstat.TH MSC 60F0562G35

keywords FréchetmeanKolmogorov-Fellerweaklawnon-compactsymmetricspacesheavy-taileddistributionsgeodesicallyprobabilityonmanifoldsHadamardrobuststatistics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a Kolmogorov-Feller weak law of large numbers for sample Fréchet means on non-compact symmetric spaces—spaces such as hyperbolic space and the space of symmetric positive-definite matrices. The main theorem says that for independent random variables that are geodesically symmetric about a common point µ, the sample Fréchet mean converges in probability to µ as long as the tail probabilities decay like the classical Kolmogorov-Feller condition; in the i.i.d. case the condition is nP(d(X_1,µ)>n) -> 0. This matters because existing laws of large numbers on manifolds require finite second moments, while the new result covers distributions with infinite first and second moments that still have moderate tails. A corollary gives a weak law under only a finite first moment, and an example verifies the non-identically distributed case on the space of positive-definite matrices.

What carries the argument

The Riemannian logarithmic map Log_µ sends each point of M into the tangent space T_µM, so the manifold can be analyzed like a vector space around µ; geodesic symmetry about µ makes the expected logarithmic map zero. Lemma 1 is the geometric inequality that carries the proof: on a Hadamard manifold (a simply connected manifold of non-positive curvature), the norm of Log_x(µ_n) is bounded above by the norm of the average of Log_x(x_i), meaning the intrinsic mean is never farther from any base point than the Euclidean average of the logarithmic images. Lemma 2 converts that inequality into the variance bound E[d²(µ_n,µ)] ≤ (1/n²)Σ E[d²(X_i,µ)] under finite variance and symmetry, which after tr

What would settle it

On the hyperbolic plane H^2, take a geodesically symmetric distribution about µ with P(d(X,µ)>t) ~ 1/(t log t) for large t and simulate sample Fréchet means for increasing sample sizes; if they fail to converge in probability to µ, Theorem 1(ii) is invalid. Alternatively, compute the population Fréchet mean of such a distribution directly and check whether it equals µ.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: on a non-compact symmetric space, if the observations are independent and each distribution is geodesically symmetric about µ, then the sample Fréchet mean µ_n converges to µ in probability under either condition (i) the sum of tail probabilities Σ P(d(X_i,µ)>n) tends to 0 and the normalized truncated second moments n^{-2}Σ E[d²(X_{n,i},µ)] tend to 0, or condition (ii) the variables are i.i.d. and nP(d(X_1,µ)>n) -> 0. The proof truncates the observations at distance n from µ, shows the truncated sample Fréchet mean has variance bounded by (1/n²) times the sum of truncated second moments, then uses the tail condition to force the truncation error and variance t

Load-bearing premise

The argument assumes that every geodesically symmetric distribution has a unique population Fréchet mean given by the center of symmetry µ, so that the logarithmic map has expectation zero; if that identification fails, the variance bound no longer points at µ and the theorem's conclusion cannot be reached.

Editorial extensions

If this is right

In the i.i.d. symmetric case, the sample Fréchet mean is consistent even when the first moment is infinite; a distribution with tail P(d(X,µ)>t) ∝ 1/(t log t) satisfies the condition.
Independent, non-identically distributed observations with a common center of symmetry also produce consistency, provided the two truncated-moment sums vanish.
If only a finite first moment is assumed, the weak law follows directly (Corollary 1).
The variance modulation m_n = nE[d²(µ_n,µ)]/E[d²(X,µ)] is at most 1, so curvature does not slow the mean-squared convergence relative to Euclidean space; under negative curvature it is strictly less than 1 unless the distribution is supported on a geodesic.
In Euclidean spaces, the i.i.d. tail condition is not only sufficient but necessary for convergence in probability, so the theorem locates the exact threshold in that case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the authors' conjecture in Theorem 2 extends to all non-compact symmetric spaces, the Kolmogorov-Feller tail condition would be necessary as well as sufficient for i.i.d. symmetric data, giving a sharp boundary for consistency.
The same truncation-plus-shrinkage mechanism may imply finite-sample concentration for µ_n without any moment assumptions, yielding confidence sets for µ in heavy-tailed settings—an application the paper does not develop.
The variance bound suggests a direct test in simulations on the hyperbolic plane: for a distribution with tail ~1/(t log t), the empirical risk of the sample Fréchet mean should approach the population risk at rate around 1/n despite the infinite mean.
Since the paper notes the classical Fréchet mean gains robustness under these tail conditions, one could probe whether robust M-estimators such as Huber means inherit the same Kolmogorov-Feller threshold; the authors list this as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: Theorem 1 is a genuinely new result and the proof is essentially sound. The paper should go to a referee, with the expectation of a modest revision.

What it does well: Prior LLNs for Fréchet means on manifolds (Ziezold, Bhattacharya–Patrangenaru, Schötz, Evans–Jaffe) all assume finite second moments. Under a geodesic-symmetry assumption on a non-compact symmetric space, the authors prove the WLLN under the Kolmogorov–Feller tail condition, allowing infinite first and second moments. The non-i.i.d. version in (i) is a nice bonus. The proof uses a clean variance inequality (Lemma 1) plus truncation; the inequality itself is a useful by-product, and Example 1 with SPD matrices is concrete.

Soft spots, in proportion: - Theorem 2's proof contains a real error. The factorization in (7) treats a union event as a single-index conditional probability, which is invalid: P(∃i A_i ∩ B_i) is not equal to P(B_j | ∃i A_i) P(∃i A_i). The claimed conditional probability bound of 1/2 is not justified for the union. This leaves the Euclidean partial converse unproven as written. It's peripheral to Theorem 1, but the authors should either fix it or restate it as a conjecture. I would not block acceptance on this alone, but it needs to be addressed. - Lemma 2 cites Lee and Jung (2024) for uniqueness of the population Fréchet mean and E[Log_µ X_i]=0. That citation is unnecessary: for the truncated variables in the proof, geodesic symmetry about µ implies Log_µ(X_{n,i}) and −Log_µ(X_{n,i}) have the same distribution, and the variables are bounded, so the expectation is zero without any uniqueness argument. The "circularity" concern is therefore minor; it's a lazy citation, not a gap. - Corollary 1's proof has a typo: it says nP(d(X_1,µ)>n) → ∞, which should be → 0. Harmless but should be fixed.

Who should read it: researchers in probability on manifolds and geometric statistics, especially those dealing with heavy-tailed data in curved spaces. Theorem 1 is the takeaway; the rest is supporting material.

Recommendation: send to peer review. The main theorem is correct, new, and clearly positioned against the literature. The flaws are localized and fixable. I would set confidence high on Theorem 1 and moderate on Theorem 2.

Referee Report

1 major / 4 minor

Summary. The paper proves a Kolmogorov-Feller weak law of large numbers for sample Fréchet means on non-compact symmetric spaces. Under a geodesic-symmetry condition about a point μ, and for independent (not necessarily identically distributed) observations, Theorem 1 gives two sufficient conditions—a Lindeberg-type condition in (i) or the classical i.i.d. tail condition nP(d(X1,μ)>n)→0 in (ii)—under which the sample Fréchet mean converges to μ in probability. The proof combines a deterministic contraction inequality for the Fréchet mean in Hadamard manifolds (Lemma 1), a variance bound (Lemma 2), and a truncation argument. Examples include the space of symmetric positive-definite matrices with increasing-variance log-normal observations, and a heavy-tailed symmetric law with infinite first moment. A partial converse (Theorem 2) is stated for Euclidean spaces.

Significance. If Theorem 1 is correct, it is a meaningful advance in geometric probability and statistics: it removes the finite second-moment assumption common in Fréchet-mean LLNs and extends the scalar Kolmogorov-Feller result to a broad class of negatively curved manifolds, including non-i.i.d. sequences. The main proof is mostly self-contained; the deterministic inequality in Lemma 1 and the truncation/variance argument in Theorem 1 are carefully presented, and the tail verification in Case II is correct. The paper also gives a variance-modulation remark that may be of independent interest. However, the proof of the Euclidean converse (Theorem 2) contains a serious algebraic error, so the paper as written requires revision. The central Theorem 1 itself appears sound.

major comments (1)

[Section 3, Proof of Theorem 2, display (6)] The lower bound on P(∃_{1≤i≤n_j} ∥X_i∥>n_j) is algebraically invalid. Let p=P(∥X_1∥>n_j). The displayed chain reads 1-{P(∥X_1∥≤n_j)}^{n_j} = [1-{1-p}]^{n_j} ≥ 1-e^{-n_j p}. But [1-{1-p}]^{n_j}=p^{n_j}, not 1-(1-p)^{n_j}, and the claimed inequality is false. The correct formula is 1-(1-p)^{n_j} ≥ 1-e^{-n_j p}. This error invalidates the contrapositive step for Theorem 2. Moreover, the subsequent conditional-probability factorization uses a free index i and equates an unconditional probability with P(⟨X_i,S_{-i}⟩≥0 | ∃ i: ∥X_i∥>n_j) without specifying i; that step is also not justified. The proof of Theorem 2 therefore needs a substantial correction, or the theorem should be restated as a conjecture.

minor comments (4)

[Section 2, Proof of Corollary 1] The display 'nP(d(X1, µ) > n) →_{n→∞} ∞' is a misprint; it should be '→_{n→∞} 0'. As written it contradicts the desired conclusion.
[Section 3, Proof of Theorem 1, Case I] The notation μ_n is overloaded: it denotes both the sample Fréchet mean of the original sample and that of the truncated sample. The inequality 'P(μ_n ≠ μ_n)' in the first display of Case I should involve two different symbols, e.g., μ̃_n and μ_n. This is confusing and should be fixed.
[Section 3, Lemma 2] The identity E[Log_μ(X_i)]=0 is delegated to Lee and Jung (2024), Proposition 2(a). Since geodesic symmetry about μ directly gives Log_μ(X_i) d= -Log_μ(X_i), and the variables under consideration have finite first moment, a short self-contained proof would remove a dependency on an arXiv preprint and make the argument clearer.
[References and typos] There are several small typographical issues: 'Comtemporary Mathematics' in the Sturm reference, 'Kolmogrov' in the Naderi et al. reference, and the earlier-noted limit in Corollary 1. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1's derivation is self-contained; the sole self-citation (Lee & Jung 2024) is redundant because geodesic symmetry alone yields E[Log_µ X_i]=0.

full rationale

The central claim, Theorem 1, is a genuine WLLN for sample Fréchet means on non-compact symmetric spaces. Its proof chain is: Lemma 1 (a deterministic contraction inequality for the sample Fréchet mean, proved in the appendix from the second variation of squared distance in Hadamard manifolds), Lemma 2 (a variance bound obtained by squaring Lemma 1 and using independence plus zero-mean log-coordinates), and finally a truncation/Chebyshev argument in Theorem 1. The only potentially circular-looking step is in the proof of Lemma 2, where the authors write: 'the population Fr´echet mean with respect to PX is unique and equals µ since M is a non-compact symmetric space (for details, see (Lee and Jung, 2024, Proposition 2(a))). Thus, for each i = 1, 2, . . . , n, it holds that E[Logµ(Xi)] = 0.' This is a self-citation, but it is not load-bearing. Under Definition 1, geodesic symmetry about µ means P_{s_µ(X)} = P_X, and on a non-compact symmetric space s_µ(x) = Exp_µ(-Log_µ(x)). Therefore Log_µ(X_i) has the same distribution as -Log_µ(X_i), and with finite variance, E[Log_µ(X_i)] = 0 follows directly from the paper's own definitions. The cited Fréchet-mean uniqueness proposition is redundant for this step, and the zero-mean property is not the theorem's conclusion. No fitted parameters, constructed predictions, or renamed inputs appear anywhere in the derivation. Theorem 2's partial converse contains a possible probabilistic gap in inequality (7), but that is a correctness concern, not circularity. Overall, the main result does not reduce to its inputs by construction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Hadamard geometry of non-compact symmetric spaces and on the geodesic symmetry condition. No free parameters are fitted to data and no new entities are postulated. The main non-standard ingredient is the identification of the population Fréchet mean, which is cited from the authors' own prior work and should be replaced by a direct proof.

assumptions (6)

domain assumption M is a k-dimensional non-compact symmetric space, hence a Hadamard manifold.
This is the setting of the paper; it guarantees the logarithmic map is a global diffeomorphism and squared distance is convex, which underlies Lemma 1 and the existence/uniqueness of sample Fréchet means (Afsari 2011).
domain assumption The distribution of each X_i is geodesically symmetric about µ, i.e., invariant under the geodesic symmetry s_µ.
This symmetry replaces the centering in the Euclidean Kolmogorov-Feller law and is needed to get E[Log_µ(X_i)] = 0 and to identify the population Fréchet mean with µ.
domain assumption The population Fréchet mean of a geodesically symmetric distribution on a non-compact symmetric space exists, is unique, and equals µ (Lee and Jung 2024, Prop 2(a)).
Used in Lemma 2 to justify E[Log_µ(X_i)] = 0; the paper cites the authors' own prior work instead of proving it, though it is a standard convexity result.
standard math Sturm's Corollary 2.5: for a transvection T of length ℓ, d(x, T(x)) ≥ ℓ.
Used in Proposition 1 to show disjointness of T^m(A) from a compact set.
standard math For a geodesic γ in a Hadamard manifold, the second derivative of squared distance satisfies f''(t) ≥ 2||γ'(t)||².
Used in Lemma 1, the key variance inequality.
standard math Probability measures on a second-countable manifold are tight.
Used in Proposition 1 to derive a contradiction.

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Pith. "Pith review of On the Kolmogorov-Feller weak law of large numbers for the Fr\'echet mean on non-compact symmetric spaces." pith.science (2026). https://pith.science/paper/WQOWOKSI

@misc{pith2026250902074,
  author       = {Pith},
  title        = {Pith review of: On the Kolmogorov-Feller weak law of large numbers for the Fr\'echet mean on non-compact symmetric spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQOWOKSI}},
  note         = {Machine review of arXiv:2509.02074}
}

read the original abstract

We prove the Kolmogorov-Feller weak law of large numbers for sample Fr\'echet means on non-compact symmetric spaces. The result covers independent, non-identically distributed data, extending beyond the i.i.d. setting. Examples of symmetric positive-definite matrices and product symmetric spaces are provided.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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" write newline "" before.all 'output.state := FUNCTION format.url url empty "" url if FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if ne...

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barticle [author] Barden , Dennis D. , Le , Huiling H. Owen , Megan M. ( 2013 ). Central limit theorems for F r \'e chet means in the space of phylogenetic trees . Electronic Journal of Probability 18 1--25 . barticle

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bbook [author] Feller , William W. ( 1971 ). An Introduction to Probability Theory and its Applications 2 . John Wiley & Sons . bbook

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