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Large Deviations Approximations to Distributions of the Total Distance of Compound Random Walks with von Mises Directions

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Abstract

This article considers the planar random walk where the direction taken by each consecutive step follows the von Mises distribution and where the number of steps of the random walk is determined by the class of inhomogeneous birth processes. Saddlepoint approximations to the distribution of the total distance covered by the random walk, i.e. of the length of the resultant vector of the individual steps, are proposed. Specific formulae are derived for the inhomogeneous Poisson process and for processes with linear contagion, which are the binomial and the negative binomial processes. A numerical example confirms the high accuracy of the proposed saddlepoint approximations.

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

    Riccardo Gatto

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  1. Riccardo Gatto
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Correspondence to Riccardo Gatto.

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Gatto, R. Large Deviations Approximations to Distributions of the Total Distance of Compound Random Walks with von Mises Directions. Methodol Comput Appl Probab 19, 843–864 (2017). https://doi.org/10.1007/s11009-016-9523-6

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  • DOI: https://doi.org/10.1007/s11009-016-9523-6

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