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.

翻译：许多科学问题涉及同时具有时间依赖性和截面依赖性的数据。尽管线性依赖性已得到广泛研究，但在非线性依赖条件下回归估计量的理论分析仍然匮乏。本研究在再生核希尔伯特空间框架内，针对非线性向量自回归模型，探讨了一种基于核的非线性动态估计方法。我们推导了正则化核估计量与非线性回归函数之间偏差的非渐近概率界。一个关键的技术贡献是在存在依赖数据的情况下，为随机矩阵的二次型建立了集中界，这一结果本身具有独立的理论价值。此外，我们刻画了多元核函数的条件，这些条件保证了最优收敛速率。