[Paper Review] Concentration for Infinitely Divisible Vectors with Independent Components

Christian Houdré, Patricia Reynaud-Bouret|ArXiv.org|Jun 29, 2006

Random Matrices and Applications7 references3 citations

TL;DR

This paper establishes dimension-free concentration inequalities for infinitely divisible random vectors with independent components by leveraging a novel covariance representation and Lévy measure analysis. It derives sharp tail bounds for Lipschitz functions via a rate function $ h_f(t) $, yielding exponential tail decay with explicit dependence on component-wise Lipschitz constants and Lévy measures, particularly improving on existing results for heavy-tailed and light-tailed distributions including Poisson and Laplace components.

ABSTRACT

For various classes of Lipschitz functions we provide dimension free concentration inequalities for infinitely divisible random vectors with independent components and finite exponential moments.

Motivation & Objective

• To establish dimension-free concentration inequalities for infinitely divisible vectors with independent components and finite exponential moments.
• To extend existing concentration results by incorporating component-wise Lipschitz constants and Lévy measures into a unified tail bound framework.
• To improve on prior results—particularly for Poisson and Laplace-distributed components—by deriving tighter, more explicit tail estimates.
• To provide a general method applicable to various classes of Lipschitz functions and Lévy measures, including bounded and unbounded support.
• To explore the interplay between the $ \ell^1 $-Lipschitz constant $ b $ and the $ \ell^2 $-Lipschitz constant $ a $ in determining concentration rates.

Proposed method

• Utilizes a covariance representation for infinitely divisible vectors based on the Lévy-Khintchine formula, decomposing expectations of products of Lipschitz functions.
• Applies a coupling argument via a bivariate Lévy measure $ z\nu_1 + (1-z)\nu_0 $ to derive a representation of the covariance $ \mathbb{E}[f(X)g(X)] - \mathbb{E}f(X)\mathbb{E}g(X) $.
• Defines a rate function $ h_f(t) = \sup_x \sum_{k=1}^d \int_\mathbb{R} |f(x+ue_k)-f(x)|^2 \frac{e^{tb_k|u|}-1}{b_k|u|} \tilde{\nu}_k(du) $, which quantifies the growth of the cumulant generating function.
• Derives the tail inequality $ \mathbb{P}(f(X) - \mathbb{E}f(X) \geq x) \leq \exp\left( -\int_0^x h_f^{-1}(s) ds \right) $, valid for $ x < h_f^{-1}(M^-) $, where $ M $ is the radius of convergence of the moment generating function.
• Applies the method to specific cases: bounded support Lévy measures, Poisson components, and Laplace components, yielding concrete bounds with explicit constants.
• Optimizes over auxiliary parameters (e.g., $ \varepsilon $) to refine tail estimates and improve rates, especially for heavy-tailed or high-dimensional settings.

Experimental results

Research questions

• RQ1 Can dimension-free concentration inequalities be established for infinitely divisible vectors with independent components under finite exponential moment conditions?
• RQ2 How do component-wise Lipschitz constants and Lévy measures jointly influence the concentration rate of Lipschitz functions?
• RQ3 Can the results be improved over existing bounds—particularly for Poisson and Laplace components—by refining the dependence on the $ \ell^1 $-Lipschitz constant $ b $?
• RQ4 What is the optimal trade-off between the $ \ell^1 $-Lipschitz constant $ b $ and the $ \ell^2 $-Lipschitz constant $ a $ in determining tail decay?
• RQ5 Can the concentration behavior be characterized uniformly across different classes of Lévy measures, including bounded and unbounded support?

Key findings

• For i.i.d. components with Laplace-type Lévy measure, the paper derives a tail bound $ \mathbb{P}(f(X) \geq f(0) + a\sqrt{d} + a\varepsilon + x) \leq \exp\left( -c \min\left( \frac{x}{b}, \frac{x^2}{ab\sqrt{d} + 2ab d / \varepsilon} \right) \right) $, showing linear decay after a $ \sqrt{d} $-order threshold.
• By optimizing over $ \varepsilon $, the paper obtains $ \mathbb{P}(f(X) \geq f(0) + a\sqrt{d} + \square a^{2/3}b^{1/3}d^{1/3}x^{1/3} + \square a^{1/2}b^{1/2}d^{1/4}x^{1/2} + \square bx) \leq e^{-x} $, which improves on previous results for heavy-tailed components.
• For i.i.d. Poisson components, the bound $ \mathbb{P}(f(X) \geq \mathbb{E}f(X) + a\sqrt{d} + x) \leq \exp\left( -C_1 \frac{x}{b} \log\left( \frac{x}{a\sqrt{d}} \right) \right) $ is shown to be tighter than prior results when $ b \ll a $ and $ a\sqrt{d} \ll \tilde{a}^2 / b $.
• The method yields a dimension-free exponential moment bound: $ \mathbb{E} \exp\left( \frac{f(X)}{bR} \log^+\left( \frac{\lambda f(X)}{bR} \right) \right) < \infty $ for $ \lambda \frac{aV^2}{bR} < 1/e $, with $ V^2 $ involving integrals of $ |u| $ and $ |u|^2 $ over the Lévy measure.
• The paper shows that replacing $ \mathbb{E}f(X) $ with $ f(0) $ in concentration bounds is feasible up to a multiplicative constant, and that the $ \ell^1 $-Lipschitz constant $ b $ can be used instead of $ a $ in moment generating function bounds, improving sharpness when $ a \gg b $.
• The framework allows for a unified treatment of various Lévy measures and provides a systematic way to derive concentration inequalities with two distinct rates—sub-Gaussian and sub-Exponential—depending on the regime of deviation.