ArcSine Law Asymptotics

Suppose two players A and B are engaged in independent repeated plays of a fair game in which each player wins or loses one unit with equal probability. The implicit symmetry of this scenario results in the counterintuitive phenomena that in a long series of plays it is not unlikely that one of the players will remain on the winning side while the other player loses for more than half of the series. This chapter derives the distribution of (a) the last time in 2m steps that a simple symmetric random walk visits zero in a finite interval, (b) the time spent on the positive side in a finite interval, and (c) the time of the last zero in a finite interval and the arcsine limit distribution for corresponding functionals of Brownian motion. The reference to first, second, and third arcsine laws largely follows nomenclature of Feller (Feller W (1968, 1971) An introduction to probability theory and its applications, vol 1, 3rd edn., vol 2, 2nd edn. Wiley, New York), commonly cited in the probability literature, although they are not derived in that order here, the first being due to Levy. Apart from its aid in illustrating an important nuance for decision makers when dealing with random phenomena, the arcsine law involves rather non-intuitive distribution of natural functionals of the random walk and Brownian motion. The asymptotic results for random walk are obtained by an application of the local limit theorem from Chapter 16. Although the functional central limit theorem of the previous Chapter 17 can also be applied, it is not required beyond identifying the random walk limits with corresponding functionals of Brownian motion.