For the symmetric random walk starting at 0:

(a) What is the expected time to return to 0?

(b) Let N, denote the number of returns by time n. Show that E[N2] = (2n+1) D) (2") ()" - 1 2n 1.

(c) Use

(b) and Stirling's approximation to show that for n large E[N] is proportional to Vn.