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

• "Twisted random walks," and

• "Brownian Motion Winding Number."

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.

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

• "Twisted random walks," and

• "Brownian Motion Winding Number."

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.