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What is Learnable in Valiant's Theory of the Learnable? 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Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus
Radomyra Shevchenko, Meryem Slaoui, Ciprian A. Tudor · 2019-03-06 · via math.ST updates on arXiv.org

We analyze the generalized $k$-variations for the solution to the wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian with Hurst parameter $H>\frac{1}{2}$ in time and which is white in space. The $k$-variations are defined along {\it filters} of any order $p\geq 1$ and of any length. We show that the sequence of generalized $k$-variation satisfies a Central Limit Theorem when $p> H+\frac{1}{4}$ and we estimate the rate of convergence for it via the Stein-Malliavin calculus. The results are applied to the estimation of the Hurst index. We construct several consistent estimators for $H$ and these estimators are analyzed theoretically and numerically.

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