  Is it possible to detect if the sample paths of a stochastic process almost
surely admit a finite expansion with respect to some/any basis? The
determination is to be made on the basis of a finite collection of
discretely/noisily observed sample paths. We show that it is indeed possible to
construct a hypothesis testing scheme that is almost surely guaranteed to make
only finitely many incorrect decisions as more data are collected. Said
differently, our scheme almost certainly detects whether the process has a
finite or infinite basis expansion for all sufficiently large sample sizes. Our
approach relies on Cover's classical test for the irrationality of a mean,
combined with tools for the non-parametric estimation of covariance operators.
