[Paper Review] Analytic and asymptotic properties of multivariate generalized Linnik's probability densities

Sungbin Lim, Lee-Peng Teo|arXiv (Cornell University)|Mar 30, 2009

Statistical Distribution Estimation and Applications47 references3 citations

TL;DR

This paper investigates the analytic and asymptotic properties of the multivariate generalized Linnik probability density $ p_{\alpha,\nu,n}(\mathbf{x}) $, whose characteristic function is $ \varphi_{\alpha,\nu,n}(\mathbf{t}) = (1 + \|\mathbf{t}\|^\alpha)^{-\nu} $. It derives integral representations and asymptotic expansions for $ p_{\alpha,\nu,n}(\mathbf{x}) $ as $ \|\mathbf{x}\| \to 0 $ and $ \|\mathbf{x}\| \to \infty $, and establishes conditions under which the density can be expressed via entire functions, particularly when $ \alpha $ belongs to a dense subset of $ (0,2) $ related to Liouville numbers.

ABSTRACT

This paper studies the properties of the probability density function $p_{α,ν, n}(\mathbf{x})$ of the $n$-variate generalized Linnik distribution whose characteristic function $φ_{α,ν,n}(\boldsymbol{t})$ is given by φ_{α,ν,n}(\boldsymbol{t})=\frac{1} {(1+\Vert\boldsymbol{t}\Vert^α)^ν}, α\in (0,2], ν>0, \boldsymbol{t}\in \mathbb{R}^n, where $\Vert\boldsymbol{t}\Vert$ is the Euclidean norm of $\boldsymbol{t}\in\mathbb{R}^n$. Integral representations of $p_{α,ν, n}(\mathbf{x})$ are obtained and used to derive the asymptotic expansions of $p_{α,ν, n}(\mathbf{x})$ when $\Vert\mathbf{x}\Vert o 0$ and $\Vert\mathbf{x}\Vert o \infty$ respectively. It is shown that under certain conditions which are arithmetic in nature, $p_{α,ν, n}(\mathbf{x})$ can be represented in terms of entire functions.

Motivation & Objective

• To study the analytic and asymptotic behavior of the multivariate generalized Linnik probability density $ p_{\alpha,\nu,n}(\mathbf{x}) $ with characteristic function $ (1 + \|\mathbf{t}\|^\alpha)^{-\nu} $.
• To derive integral representations of $ p_{\alpha,\nu,n}(\mathbf{x}) $ for use in asymptotic analysis.
• To determine the asymptotic behavior of $ p_{\alpha,\nu,n}(\mathbf{x}) $ as $ \|\mathbf{x}\| \to 0 $ and $ \|\mathbf{x}\| \to \infty $.
• To identify conditions under which $ p_{\alpha,\nu,n}(\mathbf{x}) $ can be represented as an entire function, particularly in relation to Diophantine approximation and Liouville numbers.
• To establish a connection between the convergence of series representations of the density and arithmetic properties of $ \alpha $, especially when $ \alpha\nu $ is a sum of rational and $ (2f+1) $-adic expansions.

Proposed method

• Derives integral representations of $ p_{\alpha,\nu,n}(\mathbf{x}) $ using inverse Fourier transform techniques and Bessel function identities.
• Applies asymptotic analysis to the integral representations to derive the behavior of $ p_{\alpha,\nu,n}(\mathbf{x}) $ as $ \|\mathbf{x}\| \to 0 $ and $ \|\mathbf{x}\| \to \infty $.
• Uses the dual distribution relationship: $ p_{\alpha,\nu,n}(\mathbf{x})/p_{\alpha,\nu,n}(\mathbf{0}) $ is the characteristic function of the generalized Cauchy distribution when $ \nu > n/\alpha $.
• Introduces sets $ \Omega $, $ \Lambda $, and $ E \subset (0,2) $ to characterize $ \alpha $ for which the series representations of the density diverge.
• Constructs sequences $ l_k $, $ j_k $ based on $ \alpha\nu $'s $ (2f+1) $-adic expansion to show divergence of coefficients $ A_{1,l_k} $, $ A_{2,j_k} $.
• Employs Stirling’s formula and trigonometric bounds on sine functions to estimate growth of terms in the series, proving divergence for $ z \neq 0 $.

Experimental results

Research questions

• RQ1 How does the multivariate generalized Linnik density $ p_{\alpha,\nu,n}(\mathbf{x}) $ behave asymptotically as $ \|\mathbf{x}\| \to 0 $?
• RQ2 What is the asymptotic behavior of $ p_{\alpha,\nu,n}(\mathbf{x}) $ as $ \|\mathbf{x}\| \to \infty $, and how does it depend on $ \alpha $, $ \nu $, and $ n $?
• RQ3 Under what arithmetic conditions on $ \alpha $ is $ p_{\alpha,\nu,n}(\mathbf{x}) $ representable as an entire function?
• RQ4 Can the series representations of $ p_{\alpha,\nu,n}(\mathbf{x}) $ converge for all $ \mathbf{x} $, or are there values of $ \alpha $ for which they diverge?
• RQ5 What role do Liouville-type numbers and $ (2f+1) $-adic expansions play in determining the analytic structure of the density?

Key findings

• The density $ p_{\alpha,\nu,n}(\mathbf{x}) $ admits integral representations derived from the inverse Fourier transform of its characteristic function $ (1 + \|\mathbf{t}\|^\alpha)^{-\nu} $.
• As $ \|\mathbf{x}\| \to 0 $, the density behaves like a power of $ \|\mathbf{x}\| $, with the leading-order term depending on $ \alpha $, $ \nu $, and $ n $.
• As $ \|\mathbf{x}\| \to \infty $, the density decays like $ \|\mathbf{x}\|^{-\alpha\nu - n} $, indicating heavy-tailed behavior.
• For $ \alpha \in E $, a dense subset of $ (0,2) $ defined via $ (2f+1) $-adic expansions and rational approximations, the series representations of $ p_{\alpha,\nu,n}(\mathbf{x}) $ diverge for all $ \mathbf{x} \neq \mathbf{0} $.
• The coefficients $ A_{1,l_k} $ and $ A_{2,j_k} $ in the series expansion grow without bound as $ k \to \infty $, implying divergence of the series for $ z \neq 0 $ when $ \alpha \in E $.
• The paper establishes that $ p_{\alpha,\nu,n}(\mathbf{x}) $ cannot be represented as an entire function for $ \alpha \in E $, despite $ E $ being dense in $ (0,2) $, due to divergence of the series coefficients.