PLANAR EXPONENTIAL RANDOM-WALK WITH RESTRICTED ANGLES OF TURN

The position of a target executing a random walk in a plane is analysed. The motion is at constant speed; the time on each leg is exponentially distributed, and each leg's direction is symmetrical about that of its predecessor with small turn angles more likely than large ones. The wrapped normal distribution is used to model this latter feature of the target's behaviour. The analysis is thus an extension of research by Conolly and Roberts which looked at turn angles uniformly distributed over 2-pi. Although an exact distribution of the target's position cannot be derived, it is possible to suggest approximations in the 'short term' - fewer than 100 legs. As time passes the behaviour approaches that described by Conolly and Roberts. Also studied are areas enclosing the target with a prescribed probability; these are robust and of value in formulating search plans. The analysis is relevant to studies of the long term behaviour of sums of correlated random variables.