Problem 29

Lévy and the Arc Sine Law (1939)

Problem. A fair coin is tossed successively. Let if the jth toss is a head, and if the jth toss is a tail. Define the difference between the number of heads and tails after j tosses by . Show that the probability that the last time there was an equal number of heads and tails within 2n tosses occurred at the 2kth toss is .

Solution. Let , where is the number of ways the event can be realized. We first establish an important result. We have

since the coin is fair. Using the law of total probability,

Applying the Ballot Theorem (see Problem ...