[Paper Review] Complementary upper bounds for fourth central moment with extensions and applications

Rashmi Sharma, Ravinder Kumar+2 more|arXiv (Cornell University)|Mar 12, 2015

Matrix Theory and Algorithms15 references3 citations

TL;DR

This paper establishes complementary upper bounds for the fourth central moment of a random variable confined to a finite interval, deriving inequalities that refine and extend classical results like Popoviciu's and Nagy's inequalities. It provides tight bounds for the spread of Hermitian matrices and the span of polynomials with real roots, using moment-based inequalities and applications to spectral theory and polynomial root localization.

ABSTRACT

We prove some inequalities involving fourth central moment of a random variable that takes values in a given finite interval. Both discrete and continuous cases are considered. Bounds for the spread are obtained when a given nxn complex matrix has real eigenvalues. Likewise, we discuss bounds for the spans of polynomial equations.

Motivation & Objective

• To derive new inequalities for the fourth central moment of a random variable in a bounded interval.
• To extend classical inequalities (e.g., Popoviciu, Nagy) to higher-order moments, particularly the fourth central moment.
• To apply these moment inequalities to bound the spread of Hermitian matrices and the span of polynomials with real roots.
• To improve existing estimates for matrix eigenvalue spread and polynomial root ranges using moment-based constraints.

Proposed method

• Derives an upper bound for the fourth central moment μ₄ ≤ (M−m)⁴/12 using a quadratic inequality in the centered variable.
• Establishes a refined bound involving second and third central moments: μ₄ ≤ (μ₂² − (μ₁′−m)²μ₂)/(μ₁′−m) for the upper tail and symmetric lower tail.
• Applies these moment inequalities to the eigenvalues of Hermitian matrices via the trace and variance of eigenvalues.
• Uses Newton's identities to relate power sums of polynomial roots to coefficients, enabling bounds on the span of real-rooted polynomials.
• Applies the derived moment inequalities to the roots of monic polynomials with zero trace (coefficient of xⁿ⁻¹ = 0).
• Employs optimization techniques over bounded intervals to derive tight bounds on the range of the random variable in terms of moments.

Experimental results

Research questions

• RQ1 What are tight upper bounds for the fourth central moment of a random variable restricted to a finite interval [m, M]?
• RQ2 How can the fourth central moment be bounded in terms of lower-order moments (μ₂, μ₃) and the interval bounds?
• RQ3 Can these moment inequalities be used to derive improved bounds for the spread of a Hermitian matrix?
• RQ4 How do these inequalities refine existing bounds for the span of a polynomial with real roots?
• RQ5 Can the bounds for the fourth central moment lead to better estimates for the studentized range and kurtosis-skewness relationships?

Key findings

• The paper establishes the upper bound μ₄ ≤ (M−m)⁴/12 for the fourth central moment of any random variable in [m, M], generalizing Popoviciu’s variance bound.
• For n ≥ 5, the span of a real-rooted polynomial f(x) = xⁿ + a₂xⁿ⁻² + ... satisfies (24/n(a₂² − 2a₄))¹/⁴ ≤ spn(f) ≤ 2(a₂² − 2a₄)¹/⁴.
• A tighter lower bound for the span is derived: spn(f) ≥ (432/n³(4(2−n)a₂³ − 9na₃² + 8na₂a₄))¹/⁶, improving upon Nagy’s and Popoviciu’s bounds.
• For Hermitian matrices, the bounds yield improved lower estimates for the spread: spd(A) ≥ 6.0181 and spd(A) ≥ 6.8252 from different inequalities, outperforming standard bounds.
• The matrix spread bound (3.12) gives spd(A₁) ≤ 13.559 (vs. 13.620 from (1.7)) and spd(A₂) ≤ 11.934 (vs. 12.227 from Mirsky), showing tighter estimates.
• The results provide a refined relation among skewness, kurtosis, and the studentized range, with quantitative bounds derived from moment inequalities.