  Inference for functional linear models in the presence of heteroscedastic
errors has received insufficient attention given its practical importance; in
fact, even a central limit theorem has not been studied in this case. At issue,
conditional mean estimates have complicated sampling distributions due to the
infinite dimensional regressors, where truncation bias and scaling issues are
compounded by non-constant variance under heteroscedasticity. As a foundation
for distributional inference, we establish a central limit theorem for the
estimated conditional mean under general dependent errors, and subsequently we
develop a paired bootstrap method to provide better approximations of sampling
distributions. The proposed paired bootstrap does not follow the standard
bootstrap algorithm for finite dimensional regressors, as this version fails
outside of a narrow window for implementation with functional regressors. The
reason owes to a bias with functional regressors in a naive bootstrap
construction. Our bootstrap proposal incorporates debiasing and thereby attains
much broader validity and flexibility with truncation parameters for inference
under heteroscedasticity; even when the naive approach may be valid, the
proposed bootstrap method performs better numerically. The bootstrap is applied
to construct confidence intervals for centered projections and for conducting
hypothesis tests for the multiple conditional means. Our theoretical results on
bootstrap consistency are demonstrated through simulation studies and also
illustrated with a real data example.
