[Paper Review] The conjectures of Embrechts and Goldie

Toshiro Watanabe|arXiv (Cornell University)|Nov 4, 2015

Probability and Risk Models12 references3 citations

TL;DR

This paper disproves the long-standing conjecture by Embrechts and Goldie that the class of convolution equivalent distributions (S(γ) for γ > 0) is closed under convolution roots. Using classical Wiener's approximation theorem and properties of characteristic functions, the author constructs counterexamples showing that even if a distribution's n-fold convolution lies in S(γ), the original distribution may not. The key result is that S(γ) is not closed under convolution roots for any γ > 0, resolving a three-decade-old open problem in extreme value theory and stochastic processes.

ABSTRACT

It is shown that the class of convolution equivalent distributions and the class of locally subexponential distributions are not closed under convolution roots. Moreover, two sufficient conditions for the closure under convolution roots of the class of convolution equivalent distributions are given.

Motivation & Objective

• To resolve Conjecture II of Embrechts and Goldie, which posits that the class S(γ) of convolution equivalent distributions is closed under convolution roots for γ > 0.
• To investigate whether the closure property of S(γ) under convolution roots holds, as it is crucial for limit theorems in renewal theory, queues, and Lévy processes.
• To extend the analysis to locally subexponential and Δ-subexponential classes, determining whether their closure under convolution roots holds.
• To provide a definitive counterexample to Conjecture II, settling a 30-year-old open problem in probability theory.

Proposed method

• Uses classical Wiener's approximation theorem to analyze the behavior of characteristic functions and their inverses in the context of convolution roots.
• Defines auxiliary absolutely continuous distributions μ_c with densities f_c(x) to approximate the original measure μ, preserving tail and convolution properties.
• Applies the dominated convergence theorem and asymptotic analysis of tail probabilities to derive limits involving m_c(x; {λ_k}) and M_c(x; {λ_k}) sequences.
• Employs the non-vanishing property of the characteristic function ĥ_μ(γ + iz) for all real z to ensure invertibility and uniqueness in the limit equations.
• Derives a key identity involving the integral of M_c(a - u; {λ_k}) against the (n-1)-fold convolution of μ_c, leading to a constant limit C.
• Uses Lemma 2.5 on Fourier transforms to deduce that M_c(a; {λ_k}) is constant a.e., which implies the scaling relation for tail equivalence and confirms μ_c ∈ S(γ).

Experimental results

Research questions

• RQ1 Is the class S(γ) of convolution equivalent distributions closed under convolution roots for γ > 0, as conjectured by Embrechts and Goldie?
• RQ2 Does the closure property of S(γ) under convolution roots hold for locally subexponential distributions S_loc and Δ-subexponential distributions S_Δ?
• RQ3 Can the non-closure of S(γ) under convolution roots be demonstrated via explicit counterexamples using Wiener's approximation theorem?
• RQ4 What conditions on the characteristic function and moments are necessary to ensure that μ^{n*} ∈ S(γ) implies μ ∈ S(γ)?
• RQ5 Is the class L(γ) closed under convolution roots, and how does this relate to the closure of S(γ)?

Key findings

• The class S(γ) for γ > 0 is not closed under convolution roots, thereby disproving Conjecture II of Embrechts and Goldie.
• The counterexample is constructed using a sequence of absolutely continuous approximations μ_c with densities f_c(x), ensuring that μ_c^{n*} ∈ S(γ) but μ_c ∉ S(γ) for n ≥ 2.
• For every c > 0, the distribution μ_c satisfies μ_c^{n*} ∈ S(γ), and the asymptotic tail behavior leads to a constant limit C = n^{-1} ŝ_μ_c(γ)^{1−n}, independent of the choice of sequence {λ_k}.
• The proof relies on the non-vanishing of the characteristic function ĥ_μ(γ + iz) for all real z, which allows the application of Wiener's theorem to deduce that M_c(a; {λ_k}) is constant a.e., leading to the conclusion that μ_c ∈ S(γ).
• As a consequence, the classes S_loc and S_Δ are also not closed under convolution roots, since they are subclasses of S(γ).
• The result implies that the closure property of S(γ) under convolution roots does not hold in general, even under strong moment and analyticity conditions, closing a 30-year-old open problem.