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-time problem (Brownian motion), we provide a simple alternative proof of Newman'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, taking advantage of the general set of results of Sparre Andersen. For the binomial random walk we analyse the corrections to the continuum limit on the example of the mean occupation time. The case of Cauchy-distributed steps is also studied. (author)