[Paper Review] Recent results on the stability of the parametric fundamental equation of information

Eszter Gselmann|arXiv (Cornell University)|Jul 2, 2013

Functional Equations Stability Results15 references3 citations

TL;DR

This paper establishes the Hyers–Ulam stability of the parametric fundamental equation of information for all parameters α ≠ 1 using a unified method based on functional equation techniques. It proves that α–recursive and 3–semi–symmetric information measures are stable, with explicit error bounds depending on α and initial error terms.

ABSTRACT

The purpose of this paper is to summarize the recent results on the stability of the parametric fundamental equation of information. Furthermore, by the help of a modification of a method we used in \cite{GM08} we shall give a unified proof for the Hyers--Ulam stability of the equation in question, assuming that the parameter does not equal to 1. As a corollary of the main result, a system of equations, that defines the recursive and semi--symmetric information measures is also discussed.

Motivation & Objective

• To investigate the stability of the parametric fundamental equation of information, particularly for α ≠ 1.
• To provide a unified proof of Hyers–Ulam stability for the equation using a modified method from prior work.
• To extend stability results to systems defining recursive and semi-symmetric information measures.
• To address open problems concerning stability in exceptional cases, such as α = 1 and generalized equations.
• To establish quantitative error bounds for perturbations of solutions in the context of entropy and information measures.

Proposed method

• A modified version of a functional equation method previously used in [9] is applied to prove stability for α ≠ 1.
• The proof relies on induction over the number of probabilities in the information measure, using recursive decomposition of n-tuples.
• Error bounds are derived by analyzing perturbations in the α–recursive and 3–semi–symmetric conditions, distinguishing cases by the sign of α.
• Three distinct error estimates are established: one for α < 0, one for α = 0, and one for α > 0, each based on different summation structures.
• The method uses the functional equation (1.1) as the central object, with f(x) = I₂(1−x, x) linking the equation to information measures.
• The stability analysis is conducted on the open domain D°, ensuring positivity and differentiability conditions for the involved functions.

Experimental results

Research questions

• RQ1 Is the parametric fundamental equation of information Hyers–Ulam stable for all α ≠ 1?
• RQ2 Can a unified proof technique be applied to establish stability across all α ≠ 1?
• RQ3 What is the quantitative behavior of error propagation in α–recursive and 3–semi–symmetric information measures?
• RQ4 Is the fundamental equation of information (α = 1) stable on D° or D?
• RQ5 Is the generalized fundamental equation of information of degree α stable on D° when f, g, h, k are not necessarily equal?

Key findings

• The parametric fundamental equation of information is Hyers–Ulam stable for all α ≠ 1, with a unified proof method applicable across all such α.
• For α < 0, the error bound is ∑ₖ₌₂ⁿ εₖ + K(α)(2ε₂ + ε₁)(1 + ∑ₖ₌₂ⁿ (∑ᵢ₌₁ᵏ pᵢ^α)), showing dependence on the sum of probabilities raised to α.
• For α = 0, the error bound is ∑ₖ₌₂ⁿ εₖ + K(α)n(2ε₂ + ε₁), indicating linear growth in n with respect to the number of probabilities.
• For α > 0, the same linear error bound as for α = 0 is obtained, confirming stability in the positive α regime.
• The system of α–recursive and 3–semi–symmetric information measures is stable, with error controlled by initial perturbations and the parameter α.
• The result implies that approximate solutions to the equation are uniformly close to exact solutions, supporting robustness in information-theoretic modeling.