[Paper Review] Nested subclasses of the class of $α$-selfdecomposable distributions

Makoto Maejima, Yohei Ueda|arXiv (Cornell University)|Jun 5, 2010

Stochastic processes and financial applications10 references3 citations

TL;DR

This paper introduces and analyzes nested subclasses of α-selfdecomposable distributions on ℝᵈ using two approaches: limit theorems and mappings of infinitely divisible distributions. The key contribution is a unified characterization of the limiting class of these nested subclasses as the intersection of α-selfdecomposable distributions and the closure of the class of strictly stable distributions, with explicit conditions depending on α, particularly for α ∈ (0,2).

ABSTRACT

A probability distribution $μ$ on $\mathbb R ^d$ is selfdecomposable if its characteristic function $\widehatμ(z), z\in\mathbb R ^d$, satisfies that for any $b>1$, there exists an infinitely divisible distribution $ρ_b$ satisfying $\widehatμ(z) = \widehatμ(b^{-1}z)\widehatρ_b(z)$. This concept has been generalized to the concept of $α$-selfdecomposability by many authors in the following way. Let $α\in\mathbb R$. An infinitely divisible distribution $μ$ on $\mathbb R ^d$ is $α$-selfdecomposable, if for any $b>1$, there exists an infinitely divisible distribution $ρ_b$ satisfying $\widehatμ(z) = \widehat μ(b^{-1}z)^{b^α}\widehatρ_b(z)$. By denoting the class of all $α$-selfdecomposable distributions on $\mathbb R ^d$ by $L^{\leftangleα ightangle}(\mathbb R ^d)$, we define in this paper a sequence of nested subclasses of $L^{\leftangleα ightangle}(\mathbb R ^d)$, and investigate several properties of them by two ways. One is by using limit theorems and the other is by using mappings of infinitely divisible distributions.

Motivation & Objective

• To define and study a hierarchy of nested subclasses within the class of α-selfdecomposable distributions on ℝᵈ.
• To investigate the structure and properties of these subclasses using two complementary approaches: limit theorems and mappings of infinitely divisible distributions.
• To unify the characterization of the limiting class of these nested subclasses across all real α, particularly distinguishing behavior for α ≤ 0, 0 < α < 1, α = 1, and 1 < α < 2.
• To establish a connection between the iterative application of a mapping Φₐ and the limiting class, showing convergence to a specific intersection of distribution classes.

Proposed method

• Define nested subclasses Lₘ^(⟨α⟩)(ℝᵈ) recursively by requiring that for each b > 1, the characteristic function satisfies μ̂(z) = μ̂(b⁻¹z)^{b^α} ρ_b(z), with ρ_b ∈ Lₘ₋₁^(⟨α⟩)(ℝᵈ), starting from L₀^(⟨α⟩)(ℝᵈ) = L^(⟨α⟩)(ℝᵈ).
• Use limit theorems involving infinitesimal triangular arrays: a sequence of independent random variables {Xₙ}, scaling {aₙ} ↓ 0, and centering {cₙ} such that aₙ⁻¹∑ⱼ₌₁ⁿ Xⱼ + cₙ converges in distribution to μ.
• Define a mapping Φₐ on the space of probability measures that generates the nested subclasses via iteration: Φₐ^m+1(𝕀(ℝᵈ)) generates Lₘ^(⟨α⟩)(ℝᵈ).
• Characterize the limit of these nested subclasses as m → ∞ using the intersection of the class L^(⟨α⟩)(ℝᵈ) with the closure of the class of strictly stable distributions, denoted S(ℝᵈ).
• Establish the equivalence limₘ→∞ 𝔒(Φₐ^{m+1}) = L^(⟨α⟩)(ℝᵈ) ∩ cl(S(ℝᵈ)) for all α ∈ ℝ, with refined characterizations depending on the value of α.
• Use integral conditions involving the spectral measure and the parameter β to characterize the centering term in the limit, particularly for α = 1 and α ∈ (1,2).

Experimental results

Research questions

• RQ1 What is the limiting class of the nested subclasses Lₘ^(⟨α⟩)(ℝᵈ) as m → ∞ for different values of α ∈ ℝ?
• RQ2 How do the limit theorems and mappings of infinitely divisible distributions relate in characterizing the nested subclasses of α-selfdecomposable distributions?
• RQ3 Can the limiting class of the nested subclasses be expressed uniformly across all α ∈ ℝ, and if so, what is the unified form?
• RQ4 What role does the closure of the class of strictly stable distributions play in the limit characterization of these nested subclasses?
• RQ5 How do the conditions on the Lévy measure and spectral measure affect the convergence and structure of the limiting class for different α ∈ (0,2)?

Key findings

• The limit of the nested subclasses Lₘ^(⟨α⟩)(ℝᵈ) as m → ∞ is given by L^(⟨α⟩)(ℝᵈ) ∩ cl(S(ℝᵈ)) for all α ∈ ℝ, providing a unified characterization across all real α.
• For α ≤ 0, the limit class is L^(⟨α⟩)(ℝᵈ) ∩ cl(S(ℝᵈ)) = L^(⟨α⟩)(ℝᵈ) ∩ {δ₀}, meaning the only possible limit is the degenerate distribution at zero.
• For 0 < α < 1, the limit class is L^(⟨α⟩)(ℝᵈ) ∩ cl(S(ℝᵈ)) ∩ ℂₐ(ℝᵈ), where ℂₐ(ℝᵈ) denotes a class of distributions with a specific moment condition on the spectral measure.
• For α = 1, the limit class is characterized by an additional integral condition on the spectral measure involving the Beta function and the centering term γ.
• For 1 < α < 2, the limit class is L^(⟨α⟩)(ℝᵈ) ∩ cl(S(ℝᵈ)) ∩ I₁⁰(ℝᵈ), where I₁⁰(ℝᵈ) is the class of infinitely divisible distributions with finite first absolute moment.
• The limit of the iterated mapping Φₐ^{m+1} converges to L^(⟨α⟩)(ℝᵈ) ∩ cl(S(ℝᵈ)), confirming that the mapping-based and limit-theorem-based characterizations are consistent.