[Paper Review] Absolute moments and Fourier-based probability metrics

Yong-Kum Cho|arXiv (Cornell University)|Oct 29, 2015

Geometric Analysis and Curvature Flows20 references3 citations

TL;DR

This paper introduces explicit Fourier-based formulae for computing absolute moments of probability measures on $\mathbb{R}^d$, and constructs complete probability metrics—specifically $\rho_\alpha$—that capture finite absolute moments of arbitrary order $\alpha>0$. The key contribution is the derivation of convergence rates for solutions to the heat-diffusion equation in terms of these metrics, showing $L^\infty$ decay rates proportional to $t^{-(\alpha+d+|\sigma|)/p}$, which refine classical estimates by incorporating moment information via the Fourier transform.

ABSTRACT

We present a family of explicit formulae for evaluating absolute moments of probability measures on $\mathbb{R}^d$ in terms of Fourier transforms. As to the space of probability measures possessing finite absolute moments of an arbitrary order, we exploit our formulae to characterize its Fourier image and construct Fourier-based probability metrics which make the space complete. As applications, we compute absolute moments of those probability measures whose characteristic functions belong to the Scheonberg classes, estimate absolute moments of convolutions and investigate the asymptotic behavior of solutions to the heat-diffusion equations from a probability view-point.

Motivation & Objective

• To develop a systematic framework for computing absolute moments of probability measures using Fourier transforms.
• To characterize the Fourier image of $\mathcal{P}_\alpha(\mathbb{R}^d)$, the space of probability measures with finite absolute moments of order $\alpha$.
• To construct complete, Fourier-based probability metrics sensitive to absolute moment information, overcoming limitations of the uniform metric $d_\infty$.
• To apply the framework to estimate moments of measures in Scheonberg classes and analyze the asymptotic behavior of solutions to the heat-diffusion equation.

Proposed method

• Derive explicit formulae for absolute moments of $\mu \in \mathcal{P}_\alpha(\mathbb{R}^d)$ using the Fourier transform and difference operators.
• Define the metric $\rho_\alpha(\mu, \nu) = \int_{\mathbb{R}^d} \frac{|\widehat{\mu}(\xi) - \widehat{\nu}(\xi)|}{|\xi|^{d+\alpha}} \, d\xi$ to measure distance between characteristic functions.
• Prove that $\rho_\alpha$ is a complete metric on $\mathcal{P}_\alpha(\mathbb{R}^d)$, ensuring convergence of Cauchy sequences in this space.
• Use the Fourier inversion theorem and pointwise $L^\infty$ estimates to bound derivatives of solutions to the $p$-stable Fokker-Planck equation.
• Apply calculus to maximize $|\xi|^{\alpha + d + |\sigma|} e^{-t|\xi|^p}$, yielding optimal decay constants in convergence rates.
• Establish stability estimates by relating $\|\partial^\sigma(f-g)(\cdot,t)\|_\infty$ to $\rho_\alpha(\mu, \nu)$, with explicit constant $A_\sigma = (2\pi)^{-d} \left( \frac{\alpha + d + |\sigma|}{ep} \right)^{\frac{\alpha + d + |\sigma|}{p}}$.

Experimental results

Research questions

• RQ1 How can absolute moments of probability measures on $\mathbb{R}^d$ be explicitly computed using Fourier transforms?
• RQ2 What is the Fourier image of the space $\mathcal{P}_\alpha(\mathbb{R}^d)$, and how can it be characterized?
• RQ3 Can a complete probability metric be constructed on $\mathcal{P}_\alpha(\mathbb{R}^d)$ that reflects the finiteness of absolute moments of order $\alpha$?
• RQ4 How do solutions to the heat-diffusion equation converge to their initial data in terms of moment-sensitive metrics?
• RQ5 What is the optimal rate of decay for derivatives of solutions to the $p$-stable Fokker-Planck equation in terms of initial moment information?

Key findings

• The metric $\rho_\alpha(\mu, \nu) = \int_{\mathbb{R}^d} \frac{|\widehat{\mu}(\xi) - \widehat{\nu}(\xi)|}{|\xi|^{d+\alpha}} \, d\xi$ is a complete probability metric on $\mathcal{P}_\alpha(\mathbb{R}^d)$, ensuring convergence of Cauchy sequences in this space.
• For $\mu, \nu \in \mathcal{P}_\alpha(\mathbb{R}^d)$ with $0 < \alpha < 1$, the solution difference satisfies $\|\partial^\sigma(f-g)(\cdot,t)\|_\infty \leq A_\sigma t^{-(\alpha + d + |\sigma|)/p} \rho_\alpha(\mu, \nu)$, with optimal constant $A_\sigma = (2\pi)^{-d} \left( \frac{\alpha + d + |\sigma|}{ep} \right)^{\frac{\alpha + d + |\sigma|}{p}}$.
• The convergence $f_t \to \mu$ as $t \to 0^+$ holds in the $\rho_\alpha$-metric, with $\rho_\alpha(f_t, \mu) \leq C t^{\alpha/p} \|\widehat{\mu}\|_\infty$, where $C = \frac{2\pi^{d/2} \Gamma(1 - \alpha/p)}{\alpha \Gamma(d/2)}$.
• The estimate $\|\partial^\sigma(f - E_p)(\cdot,t)\|_\infty = O\left( M_\mu(\alpha) t^{-(\alpha + d + |\sigma|)/p} \right)$ improves upon classical $L^\infty$ decay rates by incorporating moment information.
• For the heat equation ($p=2$), the bound $\|\partial^\sigma(f - G_b)(\cdot,t)\|_\infty = O\left( \left[M_\mu(\alpha) + |b|^\alpha\right] t^{-(\alpha + d + |\sigma|)/2} \right)$ holds for any center $b \in \mathbb{R}^d$, showing robustness to choice of reference solution.