Arpad filed a lawsuit in Manhattan Superem Court against Thurman, reported New York Daily News.
Thurman and Arpad got engaged in 2008 after dating for about a year. They ended their engagement twice, in 2009 and most recently in April. Luna, whose full name is Rosalind Arusha Arkadina Altalune Florence Thurman-Busson, was born in July 2012.
The actress has two other children -- Maya and Levon -- from her marriage to Ethan Hawke with whom she split in 2003, getting a divorce two years later.