This is one of those situations where I am reluctant to tell you a definite yes or no here on Avvo but my "gut reaction" is a no, you're not allowed to own a gun in Kentucky. My suggestion is that you hire an attorney in Lexington to help you with this issue. The attorney will get your criminal record and compare the Alaska statute with the relevant Kentucky statute and the Federal statute, and will be in a much better position to figure out whether you are prohibited from owning/possessing a firearm.
I am not your attorney, nor are you my client. This answer to your question is not to be considered legal advice. In all cases, I encourage you to contact a competent, licensed attorney, who will be able to consult with you, investigate the facts of your situation more thoroughly, and give you a better tailored answer. (Also, I frequently type answers on my mobile phone, so please excuse typos and autocorrect errors.)