We present a simple theory for the statistics of subsequent passages of kinks at a particular position along a fluctuating step on a crystal surface. Two situations are treated, namely one where the waiting time until the next passage is measured starting from the previous passage of a kink at the observation point, and one where the waiting time is measured starting from a random instant in time. In both cases the waiting time distribution is shown not to obey simple Poisson statistics. Monte Carlo simulations of kink motion show that Poisson statistics emerges for longer waiting times only when the creation and annihilation of kink-antikink pairs are explicitly included. Together, the theory and simulations provide an alternative interpretation of experimental results obtained by Giesen et al. with Scanning Tunneling Microscopy [M. Giesen-Seibert, H. Ibach, Surf. Sci. 316 (1994) 205].