Random Walks as Motions on the Torus

Abstract

Introducing motions with variable inclinations on the torus, we intend to treat random walks as stochastic-type motions on the torus. Compared with the probability density for sojourn time of the classical harmonic oscillator, the limiting density for random walks becomes also uniform on the torus if we define new variables. The famous aresine law can be interpreted in terms of new variables and its paradoxical feature disappears. This fact suggests that we can interprete other seemingly curious phenomena in the probability theory in terms of dynamical system.