Robust Independence Tests with Finite Sample Guarantees for Synchronous
  Stochastic Linear Systems
Ambrus Tamás Dániel Ágoston Bálint Balázs Csanád Csáji
The paper introduces robust independence tests with non-asymptotically
guaranteed significance levels for stochastic linear time-invariant systems,
assuming that the observed outputs are synchronous, which means that the
systems are driven by jointly i.i.d. noises. Our method provides bounds for the
type I error probabilities that are distribution-free, i.e., the innovations
can have arbitrary distributions. The algorithm combines confidence region
estimates with permutation tests and general dependence measures, such as the
Hilbert-Schmidt independence criterion and the distance covariance, to detect
any nonlinear dependence between the observed systems. We also prove the
consistency of our hypothesis tests under mild assumptions and demonstrate the
ideas through the example of autoregressive systems.