We consider a model called CHARME (Conditional Heteroscedastic Autoregressive Mixture of Experts), a class of generalized mixture of nonlinear nonparametric AR-ARCH times series. Under certain Lipschitz-type conditions on the autoregressive and volatility functions, we prove that this model is $\tau$-weakly dependent in the sense of Dedecker & Prieur (2004), and therefore, ergodic and stationary. This result forms the theoretical basis for deriving an asymptotic theory of the underlying nonparametric estimation. As application, for the case of a single expert, we use the universal approximation property of the neural networks in order to develop an estimation theory for the autoregressive function by deep neural networks, where the consistency of the estimator of neurons and bias are guaranteed.