High-dimensional vector autoregressive (VAR) models have numerous applications in fields such as econometrics, biology, climatology, among others. While prior research has mainly focused on linear VAR models, these approaches can be restrictive in practice. To address this, we introduce a high-dimensional non-parametric sparse additive model, providing a more flexible framework. Our method employs basis expansions to construct high-dimensional nonlinear VAR models. We derive convergence rates and model selection consistency for least squared estimators, considering dependence measures of the processes, error moment conditions, sparsity, and basis expansions. Our theory significantly extends prior linear VAR models by incorporating both non-Gaussianity and non-linearity. As a key contribution, we derive sharp Bernstein-type inequalities for tail probabilities in both non-sub-Gaussian linear and nonlinear VAR processes, which match the classical Bernstein inequality for independent random variables. Additionally, we present numerical experiments that support our theoretical findings and demonstrate the advantages of the nonlinear VAR model for a gene expression time series dataset.

翻译：高维向量自回归（VAR）模型在计量经济学、生物学、气候学等多个领域具有广泛应用。先前的研究主要集中于线性VAR模型，但这些方法在实践中可能具有局限性。为解决此问题，我们引入了一种高维非参数稀疏可加模型，提供了一个更为灵活的框架。我们的方法利用基展开来构建高维非线性VAR模型。在考虑过程的相依性度量、误差矩条件、稀疏性及基展开的基础上，我们推导了最小二乘估计量的收敛速率和模型选择一致性。我们的理论通过纳入非高斯性和非线性，显著扩展了先前的线性VAR模型。作为一项关键贡献，我们针对非亚高斯线性及非线性VAR过程的尾部概率，推导了尖锐的伯恩斯坦型不等式，该结果与独立随机变量的经典伯恩斯坦不等式相匹配。此外，我们通过数值实验支持了理论发现，并展示了非线性VAR模型在基因表达时间序列数据集上的优势。