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Advances on asymptotic normality in non-parametric functional time series analysis

We consider a stationary process and wish to predict future values from previous ones. Instead of considering the process in its discretized form, we choose to see it as a sample of dependent curves. Then, we cut the process into N successive curves. Obviously, the N curves are not independent. The prediction issue can be translated into a non-parametric functional regression problem from dependent functional variables. This paper aims to revisit and complete two recent works on this topic. This article extends recent literature and provides asymptotic law with explicit constants under α-mixing assumptions. Then we establish pointwise confidence bands for the regression function. To conclude, we present how our results behave on a simulation and on a real time series.