Statistical learning for $\psi$-weakly dependent processes

We consider statistical learning question for $\psi$-weakly dependent processes, that unifies a large class of weak dependence conditions such as mixing, association,$\cdots$ The consistency of the empirical risk minimization algorithm is established. We derive the generalization bounds and provide the learning rate, which, on some H{ö}lder class of hypothesis, is close to the usual $O(n^{-1\mathrm{/}2})$ obtained in the {\it i.i.d.} case. Application to time series prediction is carried out with an example of causal models with exogenous covariates.