We consider the fixed-design regression model with long-range dependen
t normal errors and show that the finite-dimensional distributions of
the properly normalized Gasser-Muller and Priestley-Chao estimators of
the regression function converge to those of a white noise process. F
urthermore, the distributions of the suitably renormalized maximal dev
iations over an increasingly finer grid converge to the Gumbel distrib
ution. These results contrast with our previous findings for the corre
sponding problem of estimating the marginal density of long-range depe
ndent stationary sequences.