Eh, considering this is a fantasy world I don't think it's fair to suggest a child can't fight in the Arena, though the very definition of a child is someone underdeveloped both physically and mentally, so the inherent disadvantages far outweigh the questionable advantages of a smaller stature and perhaps your opponent's principles.

Still, if you want to make the character go ahead, just don't expect it to be easy to find someone willing to fight them, I would have difficulty finding a suitable character in my repertoire willing to fight a child. 

Hell, I've barely managed to find anyone to fight any of my characters, cowards, cowards everywhere!