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arxiv: 1906.08376 · v1 · pith:GIVPVMMQnew · submitted 2019-06-19 · 🧮 math.PR · math.ST· stat.TH

On infinite covariance expansions

Marie Ernst , Gesine Reinert , Yvik Swan This is my paper

Pith reviewed 2026-05-25 19:44 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH
keywords Lagrange identityvariance expansionsPapathanasiou expansionscovariance expansionstest functionsunivariate distributionsprobabilistic representationsweights
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The pith

A probabilistic representation of Lagrange's identity produces Papathanasiou-type variance expansions of arbitrary order for any univariate distribution.

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

The paper develops a probabilistic version of Lagrange's identity. This identity is then applied to derive variance expansions that can be carried to any order. The resulting expansions use weights that depend on a freely chosen sequence of non-decreasing test functions. These expansions apply equally to continuous and discrete distributions under only weak assumptions on the target distribution.

Core claim

We provide a probabilistic representation of Lagrange's identity which we use to obtain Papathanasiou-type variance expansions of arbitrary order. Our expansions lead to generalized sequences of weights which depend on an arbitrarily chosen sequence of (non-decreasing) test functions. The expansions hold for arbitrary univariate target distribution under weak assumptions, in particular they hold for continuous and discrete distributions alike.

What carries the argument

The probabilistic representation of Lagrange's identity, which serves as the basis for constructing the arbitrary-order variance expansions and the associated weight sequences.

If this is right

  • Variance expansions of any finite order can be derived using the identity.
  • Generalized sequences of weights arise from the choice of non-decreasing test functions.
  • The expansions apply to both continuous and discrete distributions.
  • Concrete expansions exist for standard distributions including Pearson, Cauchy, and Levy families.

Where Pith is reading between the lines

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

  • The flexibility in test function choice may allow tailoring expansions to improve approximation accuracy in specific estimation problems.
  • The representation could connect to moment calculations or other identities used in Stein's method for distributional approximations.
  • Similar probabilistic identities might be sought for covariance structures in higher dimensions or for other functionals beyond variance.

Load-bearing premise

The target distribution satisfies only weak assumptions that permit the probabilistic representation of Lagrange's identity to hold, with the test functions required to be non-decreasing.

What would settle it

A specific univariate distribution meeting the weak assumptions for which the probabilistic representation of Lagrange's identity fails to yield the claimed Papathanasiou-type expansion at some finite order.

read the original abstract

In this paper we provide a probabilistic representation of Lagrange's identity which we use to obtain Papathanasiou-type variance expansions of arbitrary order. Our expansions lead to generalized sequences of weights which depend on an arbitrarily chosen sequence of (non-decreasing) test functions. The expansions hold for arbitrary univariate target distribution under weak assumptions, in particular they hold for continuous and discrete distributions alike. The weights are studied under different sets of assumptions either on the test functions or on the underlying distributions. Many concrete illustrations for standard probability distributions are provided (including Pearson, Ord, Laplace, Rayleigh, Cauchy, and Levy 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

1 major / 1 minor

Summary. The manuscript provides a probabilistic representation of Lagrange's identity and uses it to derive Papathanasiou-type variance expansions of arbitrary order. These expansions produce generalized sequences of weights depending on an arbitrarily chosen sequence of non-decreasing test functions. The results are asserted to hold for arbitrary univariate target distributions under weak assumptions, applying equally to continuous and discrete cases, with concrete illustrations for distributions including Pearson, Ord, Laplace, Rayleigh, Cauchy, and Lévy families.

Significance. If the central representation is valid under only the stated weak assumptions and the arbitrary-order expansions are rigorously derived without hidden moment or tail restrictions, the work would meaningfully extend the theory of covariance identities and variance expansions by offering a flexible, test-function-dependent framework. The explicit treatment of both continuous and discrete cases and the provision of many standard-distribution examples are positive features that would aid applicability.

major comments (1)
  1. [Main theorem / representation statement] The probabilistic representation of Lagrange's identity (the load-bearing step for all subsequent expansions) is asserted to hold at arbitrary order under weak assumptions on the target distribution and non-decreasing test functions. However, the manuscript does not supply the precise measurability, integrability, or moment conditions required for this representation to be valid for every n; if these conditions turn out to be stronger than claimed (e.g., finite moments of order n or specific tail behavior), the universality assertion for arbitrary univariate targets fails.
minor comments (1)
  1. [Abstract] The abstract supplies no derivation outline, error bounds, or verification steps for the arbitrary-order claim; a brief sketch in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the conditions for the central representation fully explicit. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The probabilistic representation of Lagrange's identity (the load-bearing step for all subsequent expansions) is asserted to hold at arbitrary order under weak assumptions on the target distribution and non-decreasing test functions. However, the manuscript does not supply the precise measurability, integrability, or moment conditions required for this representation to be valid for every n; if these conditions turn out to be stronger than claimed (e.g., finite moments of order n or specific tail behavior), the universality assertion for arbitrary univariate targets fails.

    Authors: We agree that the precise conditions must be stated explicitly to support the claim of validity under weak assumptions for arbitrary order. The manuscript currently describes the setting in terms of non-decreasing test functions and univariate target distributions but does not isolate the required measurability, integrability, and moment hypotheses that guarantee the representation at each finite n. In the revised version we will add a dedicated remark (or subsection) that lists these conditions explicitly, including any necessary integrability of the test functions against the target measure and any moment requirements that arise from the inductive construction. This will also clarify whether the stated universality holds without additional tail restrictions or whether the claim must be qualified for distributions possessing moments of all orders. revision: yes

Circularity Check

0 steps flagged

No circularity: expansions derived from external Lagrange identity plus chosen test functions

full rationale

The paper's central step is introducing a probabilistic representation of the known Lagrange identity and then constructing Papathanasiou-type expansions from it for arbitrary order, using any non-decreasing test functions. This construction is presented as a derivation rather than a tautology; the identity itself is external, the test functions are user-chosen inputs, and no fitted parameters are relabeled as predictions. No self-citation chains, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work appear in the abstract or described claims. The result is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the classical Lagrange identity (standard math) and the existence of a probabilistic representation under weak assumptions on the distribution and test functions (domain assumption). No free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Lagrange's identity admits a probabilistic representation under weak assumptions on the target distribution.
    Invoked to obtain the variance expansions of arbitrary order.
  • domain assumption Test functions are non-decreasing.
    Required for the generalized weight sequences.

pith-pipeline@v0.9.0 · 5617 in / 1288 out tokens · 20266 ms · 2026-05-25T19:44:16.058658+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 2 internal anchors

  1. [1]

    Statistics 52(2), 364–392 (2018)

    Afendras, G., Balakrishnan, N., Papadatos, N.: Orthogo nal polynomials in the cumulative Ord family and its application to variance bounds. Statistics 52(2), 364–392 (2018)

    work page 2018

  2. [2]

    Journal of Statistical Planning and Inference 141(11), 3628–3631 (2011)

    Afendras, G., Papadatos, N.: On matrix variance inequal ities. Journal of Statistical Planning and Inference 141(11), 3628–3631 (2011)

    work page 2011

  3. [3]

    Bernoulli 20(1), 245–264 (2014)

    Afendras, G., Papadatos, N.: Strengthened Chernoff-type variance bounds. Bernoulli 20(1), 245–264 (2014)

    work page 2014

  4. [4]

    Sankhy¯ a 69(2), 162–189 (2007)

    Afendras, G., Papadatos, N., Papathanasiou, V.: The dis crete Mohr and Noll inequality with applications to variance bounds. Sankhy¯ a 69(2), 162–189 (2007)

    work page 2007

  5. [5]

    Journal of Multivariate Analysis 131, 265–278 (2014)

    Bl´ azquez, F.L., Mi˜ no, B.S.: Maximal correlation in a non-diagonal case. Journal of Multivariate Analysis 131, 265–278 (2014)

    work page 2014

  6. [6]

    Bernoulli 7(3), 439–451 (2001)

    Bobkov, S.G., G¨ otze, F., Houdr´ e, C.: On Gaussian and Be rnoulli covariance representations. Bernoulli 7(3), 439–451 (2001). DOI 10.2307/3318495

    work page doi:10.2307/3318495 2001

  7. [7]

    Statistics & Probability Letters 3, 175–184 (1985)

    Cacoullos, T., Papathanasiou, V.: On upper and lower bou nds for the variance of functions of a random variable. Statistics & Probability Letters 3, 175–184 (1985)

    work page 1985

  8. [8]

    Springer (2014)

    Chatterjee, S.: Superconcentration and related topics . Springer (2014)

    work page 2014

  9. [9]

    Zeitschrift f¨ ur Wahrscheinlichkeit- stheorie und Verwandte Gebiete 69(2), 251–277 (1985)

    Chen, L.H.: Poincar´ e-type inequalities via stochastic integrals. Zeitschrift f¨ ur Wahrscheinlichkeit- stheorie und Verwandte Gebiete 69(2), 251–277 (1985)

    work page 1985

  10. [10]

    The Annals of Probability 9(3), 533–535 (1981)

    Chernoff, H.: A note on an inequality involving the normal distribution. The Annals of Probability 9(3), 533–535 (1981)

    work page 1981

  11. [11]

    Annales of the Institute Henri Poincar´ e (B) Probability and Statististics 55(2), 777–790 (2019)

    Courtade, T.A., Fathi, M., Pananjady, A.: Existence of Stein kernels under a spectral gap, and discrepancy bound. Annales of the Institute Henri Poincar´ e (B) Probability and Statististics 55(2), 777–790 (2019)

    work page 2019

  12. [12]

    Journal of Multi- variate Analysis 99(10), 2497–2507 (2008)

    Cuadras, C.M., Cuadras, D.: Eigenanalysis on a bivaria te covariance kernel. Journal of Multi- variate Analysis 99(10), 2497–2507 (2008)

    work page 2008

  13. [13]

    Electronic Journal of Probability 20, 1–28 (2015)

    Eichelsbacher, P., Th¨ ale, C.: Malliavin-Stein metho d for variance-gamma approximation on Wiener space. Electronic Journal of Probability 20, 1–28 (2015)

    work page 2015

  14. [14]

    Submitted for publication

    Ernst, M., Reinert, G., Swan, Y.: Covariance inequalit ies via Stein’s method (2019). Submitted for publication

    work page 2019

  15. [15]

    Higher-order Stein kernels for Gaussian approximation

    Fathi, M.: Higher-Order Stein kernels for Gaussian app roximation. arXiv preprint arXiv:1812.02703 (2018) 13

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  16. [16]

    arXiv preprin t arXiv:1804.04699 (2018)

    Fathi, M.: Stein kernels and moment maps. arXiv preprin t arXiv:1804.04699 (2018)

    work page arXiv 2018

  17. [17]

    ESAIM: Probability and Statistics 18, 703–712 (2014)

    Hillion, E., Johnson, O., Yu, Y.: A natural derivative o n [0,n ] and a binomial Poincar´ e inequality. ESAIM: Probability and Statistics 18, 703–712 (2014)

    work page 2014

  18. [18]

    Journal of Theoretical Probability 8(1), 23–30 (1995)

    Houdr´ e, C., Kagan, A.: Variance inequalities for func tions of Gaussian variables. Journal of Theoretical Probability 8(1), 23–30 (1995)

    work page 1995

  19. [19]

    The Annals of Probability 23(1), 400–419 (1995)

    Houdr´ e, C., P´ erez-Abreu, V.: Covariance identities and inequalities for functionals on Wiener and Poisson spaces. The Annals of Probability 23(1), 400–419 (1995)

    work page 1995

  20. [20]

    Journal of Fourier Anal ysis and Applications 4(6), 651–668 (1998)

    Houdr´ e, C., P´ erez-Abreu, V., Surgailis, D.: Interpolation, correlation identities, and inequalities for infinitely divisible variables. Journal of Fourier Anal ysis and Applications 4(6), 651–668 (1998)

    work page 1998

  21. [21]

    Statistics & Risk Modeling 11(3), 273–278 (1993)

    Johnson, R.W.: A note on variance bounds for a function o f a Pearson variate. Statistics & Risk Modeling 11(3), 273–278 (1993)

    work page 1993

  22. [22]

    The Annal s of Probability 13(3), 966–974 (1985)

    Klaassen, C.A.J.: On an inequality of Chernoff. The Annal s of Probability 13(3), 966–974 (1985)

    work page 1985

  23. [23]

    Statistics & Probability Letters 28(1), 91–97 (1996)

    Koldobsky, A., Montgomery-Smith, S.J.: Inequalities of correlation type for symmetric stable random vectors. Statistics & Probability Letters 28(1), 91–97 (1996)

    work page 1996

  24. [24]

    Annals of the Institute of Statistical Mathematics 43(2), 287–295 (1991)

    Korwar, R.: On characterizations of distributions by m ean absolute deviation and variance bounds. Annals of the Institute of Statistical Mathematics 43(2), 287–295 (1991)

    work page 1991

  25. [25]

    Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure 28(4), 435–460 (1995)

    Ledoux, M.: L’alg` ebre de Lie des gradients it´ er´ es d’ un g´ en´ erateur markovien—d´ eveloppements de moyennes et entropies. Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure 28(4), 435–460 (1995)

    work page 1995

  26. [26]

    Geometric and Functional Analysis 25(1), 256–306 (2015)

    Ledoux, M., Nourdin, I., Peccati, G.: Stein’s method, l ogarithmic Sobolev and transport inequal- ities. Geometric and Functional Analysis 25(1), 256–306 (2015)

    work page 2015

  27. [27]

    Prob- ability Surveys 14, 1–52 (2017)

    Ley, C., Reinert, G., Swan, Y.: Stein’s method for compa rison of univariate distributions. Prob- ability Surveys 14, 1–52 (2017)

    work page 2017

  28. [28]

    Miclo, L.: Quand est-ce que des bornes de Hardy permette nt de calculer une constante de Poincar´ e exacte sur la droite ? In: Annales de la Facult´ e des Sciences de Toulouse, vol. 17, pp. 121–192 (2008)

    work page 2008

  29. [29]

    Mathematische Nachrichten 7(1), 55–59 (1952)

    Mohr, E., Noll, W.: Eine Bemerkung zur Schwarzschen Ung leichheit. Mathematische Nachrichten 7(1), 55–59 (1952)

    work page 1952

  30. [30]

    Jou rnal of Statistical Planning and Inference 130(1-2), 351–358 (2005)

    Olkin, I., Shepp, L.: A matrix variance inequality. Jou rnal of Statistical Planning and Inference 130(1-2), 351–358 (2005)

    work page 2005

  31. [31]

    Statis- tics & Probability Letters 7(1), 29–33 (1988)

    Papathanasiou, V.: Variance bounds by a generalizatio n of the Cauchy-Schwarz inequality. Statis- tics & Probability Letters 7(1), 29–33 (1988)

    work page 1988

  32. [32]

    ALEA 11(2), 571–587 (2014)

    Pike, J., Ren, H.: Steins method and the Laplace distrib ution. ALEA 11(2), 571–587 (2014)

    work page 2014

  33. [33]

    Weighted Poincar\'e inequalities, concentration inequalities and tail bounds related to the Stein kernels in dimension one

    Saumard, A.: Weighted Poincar´ e inequalities, concen tration inequalities and tail bounds related to the behavior of the Stein kernel in dimension one. arXiv pr eprint arXiv:1804.03926 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  34. [34]

    I nstitute of Mathematical Statistics Lecture Notes—Monograph Series, 7

    Stein, C.: Approximate computation of expectations. I nstitute of Mathematical Statistics Lecture Notes—Monograph Series, 7. Institute of Mathematical Stat istics, Hayward, CA (1986)

    work page 1986

  35. [35]

    Tanguy, K.: Quelques in´ egalit´ es de superconcentrat ion: th´ eorie et applications. Ph.D. thesis, Universit´ e Paul Sabatier-Toulouse III (2017) 14

    work page 2017

  36. [36]

    Statis- tics & Probability Letters 79(7), 873–879 (2009) A Proofs Proof of Lemma 2.3

    Wei, Z., Zhang, X.: Covariance matrix inequalities for functions of Beta random variables. Statis- tics & Probability Letters 79(7), 873–879 (2009) A Proofs Proof of Lemma 2.3. The equivalence between (2.7) and (2.6) follows from the fac t that I[X1 <X 2] + I[X1 =X2] + I[X1 >X 2] = 1 and E [( f (X2) − f (X1) )( g(X2) − g(X1) ) I[X1 <X 2] ] = E [( f (X2) −...

    work page 2009