I believe anyone who has committed any serious crimes should not even be allowed to vote so long as it has been verified to be true. As there is such a thing as false and wrongful convictions. If they are convicted for violent crimes, fraud, felonies or sexual crimes then no they should not be allowed to run for office so long as it was a rightful conviction. If they just stole a packet of cigarettes or they were forced to commit crimes for survival then maybe so long as they have cleaned up their act and have a now respectable track record that proves that they have not and will not commit any more crimes.