[Paper Review] On closedness under convolution roots related to an infinitely divisible distribution in the distribution class L(γ)

Hui Xu, Yuebao Wang+2 more|arXiv (Cornell University)|Dec 6, 2015

Bayesian Methods and Mixture Models24 references3 citations

TL;DR

This paper investigates the closedness of distribution classes under convolution roots for infinitely divisible distributions in the class $\mathcal{L}(\gamma)$. It proves that $\mathcal{L}(\gamma)\setminus\mathcal{OS}$ and $(\mathcal{L}(\gamma)\cap\mathcal{OS})\setminus\mathcal{S}(\gamma)$ are not closed under convolution roots by constructing explicit counterexamples, thereby resolving a long-standing conjecture by Embrechts and Goldie in the negative for these classes.

ABSTRACT

We consider questions related to the well-known conjecture due to Embrechts and Goldie on the closedness of different classes of heavy- and light-tailed distributions with respect to convolution roots. We show that the class L(γ)\cap OS is not closed under convolution roots related to an infinitely divisible distribution for any γ\ge0, i.e. we provide examples of infinitely divisible distributions belonging to this class such that the corresponding Levy spectral distribution does not. We also prove a similar statement for the class (L(γ)\cap OS)\ S(γ). In order to facilitate our analysis, we explore the structural properties of some of the classes of distributions, and study some properties of the well-known transformation from a heavy-tailed distribution to a light-tailed one.

Motivation & Objective

• To resolve the conjecture by Embrechts and Goldie on the closedness of distribution classes under convolution roots for infinitely divisible distributions.
• To determine whether the class $\mathcal{L}(\gamma)\setminus\mathcal{OS}$ is closed under convolution roots for $\gamma \geq 0$.
• To examine the closedness of the class $(\mathcal{L}(\gamma)\cap\mathcal{OS})\setminus\mathcal{S}(\gamma)$ under convolution roots.
• To construct explicit counterexamples showing that the Levy spectral measure of an infinitely divisible distribution in $\mathcal{L}(\gamma)\setminus\mathcal{OS}$ does not belong to the same class.

Proposed method

• Construction of a heavy-tailed distribution $F_0$ using piecewise-defined survival functions with increasing gaps between $x_i$ and $y_i$ to violate condition (4.6) while satisfying (4.7).
• Definition of a distribution $F_1$ with piecewise constant density and survival function to satisfy $F_1 \in \mathcal{L}$ and condition (4.7), but not condition (4.6).
• Use of the transformation $\overline{F_0}(x) = \overline{F_1}(x)\mathbf{1}(x < x_1) + \sum_{i=1}^\infty \left( \overline{F_1}(x_i)\mathbf{1}(x_i \leq x < y_i) + \overline{F_1}(x)\mathbf{1}(y_i \leq x < x_{i+1}) \right)$ to define $F_0$ with $\overline{F_0}(y_n - 1)/\overline{F_0}(y_n) \to a > 1$, ensuring $F_0 \notin \mathcal{L}$.
• Verification that $F_0$ satisfies condition (4.7) but not (4.6), demonstrating that these two conditions are independent.
• Proof that $F_0$ does not satisfy condition (4.6) due to $f_1(x) = o(\overline{F_1}(x))$, while $F_1$ satisfies (4.7) and $F_1 \in \mathcal{L}$.
• Use of local distribution classes $\mathcal{L}_{\text{loc}}$, $\mathcal{S}_{\text{loc}}$, $\mathcal{OS}_{\text{loc}}$, and $\mathcal{OL}_{\text{loc}}$ to extend the analysis to local versions of the conjecture.

Experimental results

Research questions

• RQ1 Is the class $\mathcal{L}(\gamma)\setminus\mathcal{OS}$ closed under convolution roots for any $\gamma \geq 0$?
• RQ2 Does the class $(\mathcal{L}(\gamma)\cap\mathcal{OS})\setminus\mathcal{S}(\gamma)$ remain closed under convolution roots?
• RQ3 Can a distribution in $\mathcal{L}(\gamma)\setminus\mathcal{OS}$ have a Levy spectral measure not in the same class?
• RQ4 Are conditions (4.6) and (4.7) logically independent for distributions in $\mathcal{L}(\gamma)$?
• RQ5 Does the local version of the Embrechts-Goldie conjecture hold for the class $(\mathcal{L}_{\text{loc}}\cap\mathcal{OS}_{\text{loc}})\setminus\mathcal{S}_{\text{loc}}$?

Key findings

• The class $\mathcal{L}(\gamma)\setminus\mathcal{OS}$ is not closed under convolution roots for any $\gamma \geq 0$, as demonstrated by constructing an infinitely divisible distribution in this class whose Levy spectral measure is not.
• The class $(\mathcal{L}(\gamma)\cap\mathcal{OS})\setminus\mathcal{S}(\gamma)$ is also not closed under convolution roots, with a counterexample provided.
• The conditions (4.6) and (4.7) are logically independent: a distribution can satisfy one without satisfying the other, as shown by explicit constructions of $F_0$ and $F_1$.
• The distribution $F_0$ constructed satisfies $\lim_{n\to\infty} \overline{F_0}(y_n - 1)/\overline{F_0}(y_n) = a > 1$, proving $F_0 \notin \mathcal{L}$, while still satisfying condition (4.7).
• The local class $(\mathcal{L}_{\text{loc}}\cap\mathcal{OS}_{\text{loc}})\setminus\mathcal{S}_{\text{loc}}$ is not closed under convolution roots, confirming a negative result for the local version of the conjecture.
• The results provide a complete negative answer to the Embrechts-Goldie conjecture in both global and local settings for the specified distribution classes.