Asymptotic Properties For Functional Semi-parametric Regression Models With AR(1) Errors

Consider a functionl semi-parametric regression models, Y = X~Tβ+ m (T )+ε, whe(r X ,Y )are real random variable taking their values in R p×R,T takes its values in a semimetric space (E, d) which is of infinite dimension(f.r.v),βis an unknown vectorin R p, m (?) is an unknown function in R. ( X ,T )is independent of the errorε.In this paper ,consider some assumptions,we build the strong consistency of the estimator (β|∧) of the unknown vectorβand the estimator m (T|∧)of the nonparametric m (?) which in the functional semi-parametric regression model with AR(1)errors and extend the results of functional semiparametric regression model with autoregressive errors.