[dspillett]

A number of jurisdictions have legally codified definitions specifically for online harassment and this behaviour certainly counts in some of them.

I suspect that in most jurisdictions this behaviour could be of legal concern.

The other word they used, which I note you don't question, is stalking. This sort of behaviour is even more likely to infringed on laws (which may not exist, or are well enforced, in all jurisdictions but certainly do and are in some) intended to manage that more specific form of harassment.