In this paper, we perform deep neural networks for learning $\psi$-weakly dependent processes. Such weak-dependence property includes a class of weak dependence conditions such as mixing, association,$\cdots$ and the setting considered here covers many commonly used situations such as: regression estimation, time series prediction, time series classification,$\cdots$ The consistency of the empirical risk minimization algorithm in the class of deep neural networks predictors is established. We achieve the generalization bound and obtain a learning rate, which is less than $\mathcal{O}(n^{-1/\alpha})$, for all $\alpha > 2 $. Applications to binary time series classification and prediction in affine causal models with exogenous covariates are carried out. Some simulation results are provided, as well as an application to the US recession data.

翻译：本文研究了使用深度神经网络对$ψ$-弱相依过程进行学习。此类弱相依性质涵盖了一系列弱依赖条件（如混合性、关联性等），本文所考虑的设定覆盖了众多常见场景，例如：回归估计、时间序列预测、时间序列分类等。我们建立了深度神经网络预测器类中经验风险最小化算法的一致性。我们得到了泛化界，并获得了学习速率（低于$\mathcal{O}(n^{-1/\alpha})$，对于所有$\alpha > 2$）。本文还将其应用于二元时间序列分类以及含外生协变量的仿射因果模型预测。我们提供了部分模拟结果，并展示了在美国经济衰退数据上的应用。