  In this paper we develop methodology for testing relevant hypotheses about
functional time series in a tuning-free way. Instead of testing for exact
equality, for example for the equality of two mean functions from two
independent time series, we propose to test the null hypothesis of no relevant
deviation. In the two sample problem this means that an $L^2$-distance between
the two mean functions is smaller than a pre-specified threshold. For such
hypotheses self-normalization, which was introduced by Shao (2010) and Shao and
Zhang (2010) and is commonly used to avoid the estimation of nuisance
parameters, is not directly applicable. We develop new self-normalized
procedures for testing relevant hypotheses in the one sample, two sample and
change point problem and investigate their asymptotic properties. Finite sample
properties of the proposed tests are illustrated by means of a simulation study
and data examples. Our main focus is on functional time series, but extensions
to other settings are also briefly discussed.
