I can't see an AI being independent as it would need a host machine, which would need power, etc. An AI would therefore be like a baby with colossal intelligence; able to win chess games and solve complex problems, yet still physically helpless and reliant on dumb old humans for its survival. On the other hand, an AI that could control humans would be unstoppable. Certain social media platforms have been lambasted recently for fake news, for influencing their members. That would be a way for an AI to gain control without being detected.
Going to whip this out quickly, so I hope it makes sense.
So, let's say AI actually becomes intelligent and independent in thought. With a minimal digital financial investment (initial funds scavenged from fractional discarded amounts), considering it has access to all digital information (where we can only consider so much), algorithms could be generated to invest and 'earn' significant funds held by various legal digitally established companies. That wealth would grow much faster than any investment firm could ever generate.
In short order, though spread out of hundreds/thousands of virtual companies, enough wealth could be generated that the AI becomes a financial world power.
Negotiations could be held virtually with a government, which in return for a significant financial investment, will allow the build of a site within their nation (so, a random attack by X nation against future facilities means that host nation was just attacked...IOW, securing the site).
At that point, an AI held shadow company could seek out remote locations it could purchase. Another might contract the build of an ultra secure site including state of the art power supply systems. State of the art servers could be designed and companies contracted to build them. Other companies could be contracted to install them, still others generate parts...really anything, all done virtually, and all of the latest and greatest technology.
On and on, not one bit of it in person as physical labor is spread out among numerous small firms (just like military development work) to ensure operations remain secret. Part of that being, to pay those firms exorbitant funds to ensure their loyalty (and if much of it is under the table, their secrecy).
While all of this is going on, the AI continues to earn even more wealth, and further secure future political influence around the globe having access to all data.
IOW, through virtual interactions, it can secure it's physical independence, via human greed and the power of information.
K2