Kernel Estimation for Nonlinear Dynamics

Many scientific problems involve data exhibiting both temporal and cross-sectional dependencies. While linear dependencies have been extensively studied, the theoretical analysis of regression estimators under nonlinear dependencies remains scarce. This work studies a kernel-based estimation procedure for nonlinear dynamics within the reproducing kernel Hilbert space framework, focusing on nonlinear vector autoregressive models. We derive nonasymptotic probabilistic bounds on the deviation between a regularized kernel estimator and the nonlinear regression function. A key technical contribution is a concentration bound for quadratic forms of stochastic matrices in the presence of dependent data, which is of independent interest. Additionally, we characterize conditions on multivariate kernels that guarantee optimal convergence rates.