Asymptotic Results of a Nonparametric Conditional Quantile Estimator for Functional Time Series

We consider the estimation of the conditional quantile function when the covariates take values in some abstract function space. The main goal of this article is to establish the almost complete convergence and the asymptotic normality of the kernel estimator of the conditional quantile under the α-mixing assumption and on the concentration properties on small balls of the probability measure of the functional regressors. Some applications and particular cases are studied. This approach can be applied in time series analysis to the prediction and building of confidence bands. We illustrate our methodology with El Niño data.