It sounds like you're asking about Donsker's Invariance Principle. See these notes:
http://www.stat.berkeley.edu/users/peres/bmall.pdf
There is an application that looks like what you might mean here. The relevant section starts on p.43 and the example is on p.44. (These page numbers are in the numbering scheme of the document, but they are actually the 46th and 47th pages of the PDF.)