Modern Stochastics: Theory and Applications

We investigate the fractional Riemann-Liouville integral of a general continuous Gaussian process, focusing first on the functional weak convergence of suitably rescaled processes. Building on these results and recent theoretical advances, we deduce functional large deviation principles. For small times, the asymptotic behavior depends solely on the covariance of the underlying process at zero, while for large times, appropriate rescaling of the covariance is required and additional regularity assumptions must be imposed. A central aspect of our work is the explicit characterization of the reproducing kernel Hilbert spaces of the fractionally integrated processes, including concrete formulas for the related norms. As a notable special case, when the underlying Gaussian process is Gauss-Markov, the reproducing kernel Hilbert spaces and the covariance structure can be described explicitly, providing precise insights into the corresponding rate functions.

We prove a quantitative functional central limit theorem for one-hidden-layer neural networks with generic activation function. Our rates of convergence depend heavily on the smoothness of the activation function, and they range from logarithmic for nondifferentiable nonlinearities such as the ReLu to $\sqrt{n}$ for highly regular activations. Our main tools are based on functional versions of the Stein–Malliavin method; in particular, we rely on a quantitative functional central limit theorem which has been recently established by Bourguin and Campese [Electron. J. Probab. 25 (2020), 150].

The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.