Arison was in London when he sent a message on his twitter account that read:
"It's 2:30 a.m. in London and I was just woken up with great news. Welcome to the family No. 20."
Allen played five years with Boston after opening his career with Milwaukee and then going to Seattle. He won an NBA title with the Celtics in 2008.
Allen has made 2,718 shots from 3-point range, most in NBA history. He has career averages of 20 points, 4.2 rebounds and 3.6 assists in 1,148 contests.
He has been an all-star 10 times.
Allen's move to Miami, along with all other free-agent signings, cannot become official until July 11.