Being convinced of - is correct Having been convinced of - is ok too.

Nice example of a dangling participle, if I'm not mistaken! 'Having convinced' would mean that the subject (the states) had managed to convince someone or something else about this - which obviously doesn't make sense. It's the states which are convinced, so the verb which goes with it is 'to be', either stated or not: Convinced of ........., Being convinced of ........., Having been convinced of, .... Whether you see the participle 'convinced' as being an adjective or part of a passive construction, you still need the verb 'to be'. An example of 'Having convinced ...' might be: Having convinced my all friends that the ride was safe, I booked tickets for us. ie 'I' am the subject of this sentence, and I booked the tickets after I had convinced other people about the safety of the ride. This is different from your sentence above, where the word 'convinced' refers to the subject. I hope that's clear.

As GuideDogSaint says but I would add the following... To be convinced of = to believe (you have already been persuaded by some sort of evidence) To convince = to persuade