Statistics of the occupation time for a random walk in the presence of a moving boundary

We investigate the distribution of the time spent by a random walker to the right of a boundary moving with constant velocity v. For the continuous-ti me problem (Brownian motion), we provide a simple alternative proof of Newm an's recent result (Newman T J 2001 J. Phys. A: Math. Gen. 34 L89) using a method developed by Kac. We then discuss the same problem for the case of a random walk in discrete time with an arbitrary distribution of steps, taki ng advantage of the general set of results of Sparre Andersen. For the bino mial random walk we analyse the corrections to the continuum limit on the e xample of the mean occupation time. The case of Cauchy-distributed steps is also studied.