KPSS test for functional time series

We develop extensions of the KPSS test to functional time series (FTS). The null hypothesis of the KPSS test for scalar data is that the series follows the model xt=α+βt+ηt, where {ηt} is a stationary time series. The alternative is the model that includes a random walk: xt=α+βt+∑i≤tui+ηt. A FTS is a collection of curves observed consecutively over time. Examples include intraday price curves, term structure curves, and intraday volatility curves. We define the relevant testing problem for FTS, formulate the required assumptions, derive test statistics, and their asymptotic distributions. These distributions are used to construct effective tests, both Monte Carlo and pivotal, which are applied to series of yield curves and examined in a simulation study.