On the probability distribution of a moving target. Asymptotic and non-asymptotic results

The problem addressed here is the probability distribution of the position of a moving target, and especially of its distance to the starting point. The trajectory is made of leg segments with random length and random change of direction, and it is assumed that the target has a known constant velocity. Earlier results have been obtained in the literature in the simple case where the change of direction is uniformly distributed on the circle and the length of leg is exponentially distributed. These results are generalized for an arbitrary (non-necessarily uniformly distributed) change of direction and an arbitrary (non-necessarily exponentially distributed) length of leg. Explicit expressions are obtained for the non-asymptotic mean and covariance matrix of the position, and a central limit theorem is obtained for the normalized position, with an explicit expression for the asymptotic variance, hence a limiting Rayleigh distribution for the normalized distance to the starting point.