There is a bunch of such statements which can be obtained by [Stein's method][1].

You might be interested in the seminal  [paper][2] by Stein. Have a look also at the very recent book [*"Normal Approximation by Stein's Method"*][3] by Chen, Goldstein and Shao.


  [1]: http://en.wikipedia.org/wiki/Stein%27s_method
  [2]: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bsmsp/1200514239
  [3]: http://books.google.co.uk/books?id=5jMVpAbs9UkC&printsec=frontcover&dq=Normal+Approximation+by+Stein%2527s+Method&hl=en&ei=UFbsTPL-FcShOqrIzJEB&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false