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math.ST updates on arXiv.org

What is Learnable in Valiant's Theory of the Learnable? 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Sparse Estimation for High-Dimensional Lévy-driven Ornstein--Uhlenbeck Processes from Discrete Observations
Niklas Dexheimer, Natalia Jeszka · 2026-03-06 · via math.ST updates on arXiv.org

We study high-dimensional drift estimation for Lévy-driven Ornstein--Uhlenbeck processes based on discrete observations. Assuming sparsity of the drift matrix, we analyze Lasso and Slope estimators constructed from approximate likelihoods and derive sharp nonasymptotic oracle inequalities. Our bounds disentangle the contributions of discretization error and stochastic fluctuations, and establish minimax optimal convergence rates under suitable choices of tuning parameters in a high-frequency regime. We further quantify the sample complexity required to attain these rates depending on the Lévy noise. The results extend the theory of high-dimensional statistics for stochastic processes to a substantially broader class of noise mechanisms, in particular pure jump processes. They also demonstrate that Lasso and Slope remain competitive for jump-driven systems, providing practical guidance for inference in applications where Lévy processes are a natural modeling choice.

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