On the consistency of bootstrap methods in separable Hilbert spaces

Hilbert spaces are frequently used in statistics as a framework to deal with general random elements, specially with functional-valued random variables. The scarcity of common parametric distribution models in this context makes it important to develop non-parametric techniques, and among them, bootstrap has already proved to be specially valuable. The aim is to establish a methodology to derive consistency results for some usual bootstrap methods when working in separable Hilbert spaces. Naive bootstrap, bootstrap with arbitrary sample size, wild bootstrap, and more generally, weighted bootstrap methods, including double bootstrap and bootstrap generated by deterministic weights with the particular case of delete −h jackknife, will be proved to be consistent by applying the proposed methodology. The main results concern the bootstrapped sample mean, however since many usual statistics can be written in terms of means by considering suitable spaces, the applicability is notable. An illustration to show how to employ the approach in the context of a functional regression problem is included.