Approximation of heavy-tailed distributions via stable-driven SDEs

2020/07/05 by Lu‐Jing Huang, Mateusz B. Majka, Huang, Lu-Jing +3 · 1 citation
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2007.02212

openalex publication_date 2020/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this paper we provide a rigorous theoretical framework for studying the problem of approximating heavy-tailed distributions via ergodic SDEs driven by symmetric (rotationally invariant) α-stable processes.

Citations

• Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise
• Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for\n Non-Convex Optimization
• Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm
• Quantitative Harris type theorems for diffusions and McKean-Vlasov\n processes
• On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness

Cited by

• Lévy Langevin Monte Carlo for sampling from heavy-tailed target distributions

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