Concentration Analysis of Multivariate Elliptic Diffusions
Lukas Trottner, Cathrine Aeckerle-Willems, Claudia Strauch; 24(106):1−38, 2023.
Abstract
This study presents concentration inequalities and associated PAC bounds for both continuous- and discrete-time additive functionals of multivariate, nonreversible diffusion processes. The analysis is based on an approach using the Poisson equation, allowing for the consideration of a wide range of subexponentially ergodic, multivariate diffusion processes, even those that are unbounded. These results expand on existing concentration inequalities for additive functionals of diffusion processes, which were previously only available for bounded functions or for unbounded functions of processes from a smaller class. The power of these exponential inequalities is demonstrated through two examples in different areas. Firstly, in the context of a possibly high-dimensional, parametric, nonlinear drift model under sparsity constraints, the continuous-time concentration results are used to validate the restricted eigenvalue condition for Lasso estimation, which is crucial for deriving oracle inequalities. Secondly, the results for discrete additive functionals are applied to investigate the unadjusted Langevin MCMC algorithm for sampling moderately heavy-tailed densities π. In particular, PAC bounds are provided for the sample Monte Carlo estimator of integrals π(f) for polynomially growing functions f, which quantify the necessary sample and step sizes for achieving approximation within a prescribed margin with high probability.
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