[Paper Review] Limit Theorems For Sequences of Tempered Stable and Related Distributions

Michael Grabchak|arXiv (Cornell University)|Jan 29, 2012

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper introduces the class of extended $p$-tempered $\alpha$-stable distributions ($ETS^{p}_{\alpha}$) as the closure of $p$-tempered $\alpha$-stable distributions under weak convergence. It establishes necessary and sufficient conditions for weak convergence within this class and proves that any $d$-dimensional $ETS^{p}_{\alpha}$ distribution can be approximated by linear combinations of elementary $p$-tempered $\alpha$-stable random variables, enabling simulation via discrete sums.

ABSTRACT

In this paper we define the closure under weak convergence of the class of p-tempered α-stable distributions. We give necessary and sufficient conditions for convergence of sequences in this class. Moreover, we show that any element in this class can be approximated by the distribution of a linear combination of elementary p-tempered α-stable random variables.

Motivation & Objective

• To define and characterize the smallest class closed under weak convergence that contains $p$-tempered $\alpha$-stable distributions.
• To identify necessary and sufficient conditions for weak convergence of sequences in the $TS^{p}_{\alpha}$ class.
• To show that every $d$-dimensional $ETS^{p}_{\alpha}$ distribution can be approximated by finite linear combinations of elementary $p$-tempered $\alpha$-stable random vectors.
• To extend the class $TS^{p}_{\alpha}$ to $ETS^{p}_{\alpha}$ to include limits with Gaussian components and non-decaying tail behavior.

Proposed method

• Define $ETS^{p}_{\alpha}$ as the closure of $TS^{p}_{\alpha}$ under weak convergence, allowing Gaussian components and relaxing the decay condition on the Rosiński measure.
• Use a compactification of $\mathbb{R}^d$ to analyze the behavior of Lévy measures at infinity and at zero.
• Construct a measure $\nu$ on the compactified space $\bar{\mathbb{R}}^d$ to represent the extended Rosiński measure of $ETS^{p}_{\alpha}$ distributions.
• Apply the Portmanteau Theorem and dominated convergence to establish weak convergence of approximating sequences $\mu_n$ to $\mu$.
• Use the integral representation of characteristic functions via Lévy triplets to analyze convergence in distribution.
• Prove that any $ETS^{p}_{\alpha}$ distribution arises as the weak limit of sums of independent elementary $p$-tempered $\alpha$-stable random vectors.

Experimental results

Research questions

• RQ1 What is the smallest class of distributions that contains $TS^{p}_{\alpha}$ and is closed under weak convergence?
• RQ2 What are the necessary and sufficient conditions for a sequence of $TS^{p}_{\alpha}$ distributions to converge weakly?
• RQ3 Can every $d$-dimensional $ETS^{p}_{\alpha}$ distribution be approximated by linear combinations of elementary $p$-tempered $\alpha$-stable random variables?
• RQ4 How does the extended class $ETS^{p}_{\alpha}$ differ from the class $J_{\alpha,p}$ when $\alpha \in (0,2)$?
• RQ5 What role does the Rosiński measure play in characterizing weak limits of $TS^{p}_{\alpha}$ sequences?

Key findings

• The class $ETS^{p}_{\alpha}$ is the closure of $TS^{p}_{\alpha}$ under weak convergence, and it properly contains $J_{\alpha,p}$ when $\alpha \in (0,2)$.
• Weak convergence of $TS^{p}_{\alpha}$ sequences is characterized by convergence of their Rosiński measures on the compactified space $\bar{\mathbb{R}}^d$.
• Any $d$-dimensional $ETS^{p}_{\alpha}$ distribution can be approximated by the distribution of a finite sum of independent elementary $p$-tempered $\alpha$-stable random vectors.
• The extended class $ETS^{p}_{\alpha}$ includes distributions with Gaussian components and allows for non-vanishing mass at infinity in the Rosiński measure.
• The approximation result enables simulation of $ETS^{p}_{\alpha}$ random vectors via sums of i.i.d. elementary $p$-tempered $\alpha$-stable variables.
• For $\alpha \leq 0$, $ETS^{p}_{\alpha}$ coincides with $J_{\alpha,p}$, but for $\alpha \in (0,2)$, $J_{\alpha,p} \subsetneq ETS^{p}_{\alpha}$.