These two earlier MO questions should shed light on your question: 

 - "[Twisted random walks](http://mathoverflow.net/q/138680/6094)," and 

 - "[Brownian Motion Winding Number](http://mathoverflow.net/q/42069/6094)."

The former question was answered by Andreas Rüdinger with several relevant references, and this summary: "the root mean square winding number grows logarithmically" with the number of steps.