  We investigate the nonparametric bivariate additive regression estimation in
the random design and long-memory errors and construct adaptive thresholding
estimators based on wavelet series. The proposed approach achieves
asymptotically near-optimal convergence rates when the unknown function and its
univariate additive components belong to Besov space. We consider the problem
under two noise structures; (1) homoskedastic Gaussian long memory errors and
(2) heteroskedastic Gaussian long memory errors. In the homoskedastic
long-memory error case, the estimator is completely adaptive with respect to
the long-memory parameter. In the heteroskedastic long-memory case, the
estimator may not be adaptive with respect to the long-memory parameter unless
the heteroskedasticity is of polynomial form. In either case, the convergence
rates depend on the long-memory parameter only when long-memory is strong
enough, otherwise, the rates are identical to those under i.i.d. errors. The
proposed approach is extended to the general $r$-dimensional additive case,
with $r>2$, and the corresponding convergence rates are free from the curse of
dimensionality.
