pith. sign in

arxiv: 1907.07869 · v1 · pith:46OAP6TFnew · submitted 2019-07-18 · 🧮 math.ST · stat.TH

Bounds on Spreads of Matrices related to Fourth Central Moment. II

R.Sharma , R.Kumar , R.Saini , P.Devi This is my paper

Pith reviewed 2026-05-24 19:47 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords central momentseigenvalue boundsmatrix spreadpolynomial rootsinequalitiesdiscrete distributionscontinuous distributions
0
0 comments X

The pith

Inequalities on the first four central moments bound the spread of real-eigenvalue matrices and the roots of polynomials.

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

The paper derives inequalities involving the first four central moments for both discrete and continuous distributions. These inequalities are applied to obtain explicit bounds on the eigenvalues and on the spread of any matrix whose eigenvalues are all real. The same moment relations are used to bound the roots and the span of a polynomial equation. A reader would care because the results give direct estimates of eigenvalue ranges and root intervals from moment information alone, without needing to solve the characteristic equation or compute all moments explicitly.

Core claim

We derive some inequalities involving first four central moments of discrete and continuous distributions. Bounds for the eigenvalues and spread of a matrix are obtained when all its eigenvalues are real. Likewise, we discuss bounds for the roots and span of a polynomial equation.

What carries the argument

The spread of a matrix (difference between largest and smallest real eigenvalue), bounded via inequalities that relate it to the fourth central moment.

If this is right

  • Finite fourth central moment implies an upper bound on matrix spread.
  • The same moment inequalities supply bounds on the distance between the largest and smallest roots of a polynomial.
  • The derived bounds apply uniformly to both discrete and continuous distributions.
  • Eigenvalue bounds hold without requiring the matrix to be symmetric or normal, only that all eigenvalues are real.

Where Pith is reading between the lines

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

  • The moment-to-spread relation could be tested numerically on random matrices with forced real spectra.
  • In statistics the bounds might tighten moment-based estimates of distribution support width.
  • The polynomial-root version suggests a moment method for isolating real roots without Sturm sequences.

Load-bearing premise

The distributions have finite first four central moments and the matrices have exclusively real eigenvalues.

What would settle it

A single matrix with all real eigenvalues whose observed spread exceeds the numerical upper bound stated in terms of its fourth central moment would falsify the bound.

read the original abstract

We derive some inequalities involving first four central moments of discrete and continuous distributions. Bounds for the eigenvalues and spread of a matrix are obtained when all its eigenvalues are real. Likewise, we discuss bounds for the roots and span of a polynomial equation.

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 / 0 minor

Summary. The manuscript derives inequalities involving the first four central moments for discrete and continuous distributions. It obtains bounds on the eigenvalues and spread of matrices whose eigenvalues are all real, and discusses analogous bounds on the roots and span of polynomial equations.

Significance. If the derivations hold, the results would supply explicit moment-based bounds usable in statistics and matrix analysis under the stated assumptions of finite fourth moments and real eigenvalues/roots. The work is a direct continuation (part II) of prior bounds, but its incremental contribution cannot be assessed without the explicit inequalities or proofs.

major comments (1)
  1. No derivations, explicit inequalities, or proofs are supplied in the provided manuscript text. The central claims (moment inequalities and eigenvalue bounds) cannot be verified for correctness or load-bearing assumptions such as the precise conditions under which the bounds are attained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and comments on the manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: No derivations, explicit inequalities, or proofs are supplied in the provided manuscript text. The central claims (moment inequalities and eigenvalue bounds) cannot be verified for correctness or load-bearing assumptions such as the precise conditions under which the bounds are attained.

    Authors: We agree that the provided manuscript text consists only of the abstract and does not include the derivations, explicit inequalities, or proofs. We will revise the manuscript to incorporate the full derivations of the inequalities involving the first four central moments, the explicit bounds on eigenvalues/spread for real-eigenvalue matrices, and the analogous results for polynomial roots, along with clear statements of all assumptions and conditions for attainment of the bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract states that inequalities involving the first four central moments are derived and that bounds on eigenvalues/spreads (for real-eigenvalue matrices) and polynomial roots are obtained under the explicit prerequisites of finite fourth moments and exclusively real eigenvalues/roots. No equations, fitted parameters, self-citations, or derivation steps appear in the supplied text. The work is presented as direct derivation under stated conditions rather than an unconditional claim or a result that reduces to its own inputs by construction. No load-bearing self-referential steps are detectable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient detail to identify free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5555 in / 925 out tokens · 18551 ms · 2026-05-24T19:47:34.902332+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. Reliable high-accuracy error mitigation for utility-scale quantum circuits

    quant-ph 2025-08 conditional novelty 6.0

    QESEM is a characterization-based error mitigation technique that achieves unbiased estimates with substantially reduced runtime cost compared to probabilistic error cancellation while outperforming zero-noise extrapo...

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · cited by 1 Pith paper

  1. [1]

    S.: A treatment of the Cauc hy-Schwarz inequality in C∗-modules

    Arambasic, L., Bakic, D., Moslehian, M. S.: A treatment of the Cauc hy-Schwarz inequality in C∗-modules. J. Math. Anal. Appl. 381, 546-556 (2011)

    work page 2011

  2. [2]

    Bhatia, R.: Matrix Analysis, Springer, New York (1997)

    work page 1997

  3. [3]

    Bhatia, R., Davis, C.: A better bound on the variance, Amer. Math . Monthly 107, 353-357 (2000)

    work page 2000

  4. [4]

    Bhatia, R.: Positive Definite Matrices, Princeton University Press , USA (2007)

    work page 2007

  5. [5]

    436, 1562-1571 (2012)

    Bhatia, R., Sharma, R.: Some inequalities for positive linear maps, Lin ear Algebra Appl. 436, 1562-1571 (2012)

    work page 2012

  6. [6]

    Bhatia, R., Sharma, R.: Positive linear maps and spreads of matrice s, Amer. Math. Monthly 121, 619-624 (2014)

    work page 2014

  7. [7]

    D.: Note on two papers of K.R

    Brunk, H. D.: Note on two papers of K.R. Nair, J. Indian Soc. Agric ultural Statist. 11, 186-189 (1959)

    work page 1959

  8. [8]

    R., Kumar, R., Wolkowicz, H.: Lower bounds for the sp read of a matrix, Linear Algebra Appl

    Johnson, C. R., Kumar, R., Wolkowicz, H.: Lower bounds for the sp read of a matrix, Linear Algebra Appl. 29, 161-173 (1985)

    work page 1985

  9. [9]

    496, 246-261 (2016)

    Leka, Z.: Some inequalities for central moments of matrices, Line ar Algebra Appl. 496, 246-261 (2016)

    work page 2016

  10. [10]

    K., Kumar, R.: Characterization and lower bounds f or the spread of a normal matrix, Linear Algebra Appl

    Merikoski, J. K., Kumar, R.: Characterization and lower bounds f or the spread of a normal matrix, Linear Algebra Appl. 364, 13-31 (2003)

    work page 2003

  11. [11]

    Mirsky, L.: The spread of a matrix, Mathematika 3, 127-130 (1956)

    work page 1956

  12. [12]

    Mirsky, L.: Inequalities for normal and Hermitian matrices, Duke Math. J. 24, 591-598 (1957)

    work page 1957

  13. [13]

    Murthy and V.K

    Muilwijk, J.: Note on a theorem of M.N. Murthy and V.K. Sethi, Sank hya, Ser. B 28, 183 (1966)

    work page 1966

  14. [14]

    Pearson, K.: Mathematical contribution to the theory of evolu tion, XIX; second supplement to a memoir on skew variation, Philos. Trans. Roy. Soc. London, Ser. A 216, 429-457 (1916)

    work page 1916

  15. [15]

    Popoviciu, T.: Sur les ´ equations alg´ ebriques ayant toutes leurs racines r´ eelles, Mathematica9, 129-145 (1935)

    work page 1935

  16. [16]

    M.: On the span of derivatives of polynomials, Amer

    Robinson, R. M.: On the span of derivatives of polynomials, Amer. Math. Monthly 67, 504-508 (1964)

    work page 1964

  17. [17]

    A.: How deviant can you be?, J

    Samuelson P. A.: How deviant can you be?, J. Amer. Statist. Ass oc. 63, 1522-1525 (1968)

    work page 1968

  18. [18]

    Sharma, R., Gupta, M., Kapoor, G.: Some better bounds on the v ariance with applications, J. Math. Inequal. 4, 355-363 (2010)

    work page 2010

  19. [19]

    Sharma, R., Bhandari, R., Gupta, M.: Inequalities related to the C auchy-Schwarz inequality, Sankhya 74(A), 101-111 (2012)

    work page 2012

  20. [20]

    438, 4459-4362 (2013)

    Sharma, R., Kumar, R.: Remark on upper bound for the spread o f a matrix, Linear Algebra Appl. 438, 4459-4362 (2013)

    work page 2013

  21. [21]

    Indian Acad

    Sharma, R., Saini, R.: Generalization of Samuelson’s inequality and lo cation of eigenvalues, Proc. Indian Acad. Sci. (Math. Sci.) 125, 103-111 (2015)

    work page 2015

  22. [22]

    Sci., 41, 136-137 (2016)

    Sharma, R., Saini, R.: An identity involving fourth central moment , The Math. Sci., 41, 136-137 (2016)

    work page 2016

  23. [23]

    528, 62-75 (2017)

    Sharma, R., Devi, P., Kumari, R.: A note on inequalities for positive lin ear maps, Linear Algebra Appl. 528, 62-75 (2017)

    work page 2017

  24. [24]

    Sharma, R., Kumar, R., Saini, R.,Kapoor, G.: Bounds on spreads of matrices related to fourth central moment, Bull. Malays. Math. Sci. Soc. 41, 175-190 (2018)

    work page 2018

  25. [25]

    29, 471-506 (1980)

    Wolkowicz, H., Styan, G.P.H.: Bounds for eigenvalues using traces , Linear Algebra Appl. 29, 471-506 (1980)

    work page 1980