CONVERGENCE NEAR SADDLE POINTS 3 0 x Markov Process

x+τ (x) for some λ, μ > 0 and τ (x) = O(|x|). Here the processes are indexed so that the variance of the fluctuations of X t is inversely proportional toN . The simplest example arises from the OK Corral gunfight model which was formulated byWilliams and McIlroy (1998) and studied by Kingman (1999). These processes exhibit their most interesting behaviour at times of order logN so it is necessary to establish a fluid limit that is valid for large times. We find that this limit is inherently random and obtain its distribution. Using this, it is possible to derive scaling limits for the points where these processes hit straight lines through the origin, and the minimum distance from the origin that they can attain. The power of N that gives the appropriate scaling is surprising. For example if T is the time that X t first hits one of the lines y = x or y = −x, then