8.1.1 Consider the simple random walk Sn D 1 C


Cn; S0 D 0;

in which the summands are independent with Prf D 1g D 1 2 . In Chapter 3, Section 3.5.3, we showed that the mean time for the random walk to first reach

????a < 0 or b > 0 is ab. Use this together with the invariance principle to show that E[T] D ab, where T D Ta;b D minft  0IB.t/ D ????a or B.t/ D bg;

and B.t/ is standard Brownian motion.

Hint: The approximate Brownian motion (8.11) rescales the random walk in both time and space.