Saddlepoint approximation to the distribution of the total distance of the von Mises-Fisher continuous time random walk

This article considers the random walk over Rp, with any p ≥ 2, where a particle starts at the origin and progresses stepwise with fixed step lengths and von Mises-Fisher distributed step directions. The total number of steps follows a continuous time counting process. The saddlepoint approximation to the distribution of the distance between the origin and the position of the particle at any time is derived. Despite the p-dimensionality of the random walk, the computation of the proposed saddlepoint approximation is one-dimensional and thus simple. The high accuracy of the saddlepoint approximation is illustrated by a numerical comparison with Monte Carlo simulation.