Eur. Phys. J. B (2017) 90: 238
https://doi.org/10.1140/epjb/e2017-80228-y

Regular Article

Saddlepoint approximation to the distribution of the total distance of the continuous time random walk^★

Institute of Mathematical Statistics and Actuarial Science, Department of Mathematics and Statistics, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland

^a e-mail: gatto@stat.unibe.ch

Received: 19 April 2017
Received in final form: 28 July 2017
Published online: 11 December 2017

Abstract

This article considers the random walk over R^p, with p ≥ 2, where a given particle starts at the origin and moves stepwise with uniformly distributed step directions and step lengths following a common distribution. Step directions and step lengths are independent. The case where the number of steps of the particle is fixed and the more general case where it follows an independent continuous time inhomogeneous counting process are considered. Saddlepoint approximations to the distribution of the distance from the position of the particle to the origin are provided. Despite the p-dimensional nature of the random walk, the computations of the saddlepoint approximations are one-dimensional and thus simple. Explicit formulae are derived with dimension p = 3: for uniformly and exponentially distributed step lengths, for fixed and for Poisson distributed number of steps. In these situations, the high accuracy of the saddlepoint approximations is illustrated by numerical comparisons with Monte Carlo simulation.