Interesting proofs, though I would love more of the examples that show the dataset and the unexpected results, as well as describing what may lead to these traps and how to avoid them
... if I get it right the whole idea of clustering sliding windows is wrong, the question of "what you should do instead?" is an interesting one.
I'd imagine two answers are: (1) for time series which are somewhat periodic you might cut out individual days or weeks and try to cluster them for each other, (2) for time series which are intermittent you might create some definition of an "event" (an earthquake, or a particle passing through a detector) and then cluster events and maybe (3) for something episodic such as "heart rate during a workout" you would cluster episodes.
