Here's a recent example in the Russian game Escape from Tarkov[1]. A player kills a developer and gets banned moments after for cheating. Luckily he manages to catch it on video, and posts it to reddit. He's subsequently unbanned. Why was he banned? "false alarm triggered". When asked for more details, BSG responds "Sorry, can’t tell you cause it’s part of anti cheat system, hope you understand". Which is ridiculous.
BSG is in Russia and therefore untouchable, but I don't have much hope for these other companies like ATVI who are doing the sort of abuse.
[1]. https://www.reddit.com/r/EscapefromTarkov/comments/zawwc5/i_...
This is how all fraud detection works. Outside of gaming as well. That doesn't mean the previous action was right, but the fact they can't tell you why means nothing at all. The whole industry is built on secrecy and having a few hidden tricks in the bag.
The banned guy can contact battleye and try to get unbanned if it was false, has happened before (usually a group of players with something in common are banned, last time it was s specific motherboard or a driver for it).
If EFT bans a cheater, they're effectively getting another sale, because the cheater will come back and make money off the grey market once more. The result is that you have tons of cheaters, and EFT only needs to make broad strokes efforts at curbing them to keep existing players happy.