Deep learning for ψ-weakly dependent processes

In this paper, we perform deep neural networks for learning stationary $\psi$-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 an asymptotic learning rate, which is less than $O\left(n^{-1/\alpha }\right)$, for all $\alpha >2$. A bound of the excess risk, for a wide class of target functions, is also derived. 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.