In this paper, we consider a model called CHARME (Conditional Heteroscedastic
Autoregressive Mixture of Experts), a class of generalized mixture of nonlinear
nonparametric AR-ARCH time series. Under certain Lipschitz-type conditions on
the autoregressive and volatility functions, we prove that this model is
stationary, ergodic and τ-weakly dependent. These conditions are much
weaker than those presented in the literature that treats this model. Moreover,
this result forms the theoretical basis for deriving an asymptotic theory of
the underlying (non)parametric estimation, which we present for this model. As
an application, from the universal approximation property of neural networks
(NN), we develop a learning theory for the NN-based autoregressive functions of
the model, where the strong consistency and asymptotic normality of the
considered estimator of the NN weights and biases are guaranteed under weak
conditions.Comment: 31 pages, 4 figure