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A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths

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Abstract

A constrained diffusive random walk of n steps in ℝd and a random flight in ℝd, which are equivalent, were investigated independently in recent papers (J. Stat. Phys. 127:813, 2007; J. Theor. Probab. 20:769, 2007, and J. Stat. Phys. 131:1039, 2008). The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1,2,4. Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D.

The previous walk is generalized by considering step lengths which have independent and identical gamma distributions with a shape parameter q>0. Given the total walk length being equal to 1, the step lengths have a Dirichlet distribution whose parameters are all equal to q. The walk and the flight above correspond to q=1. Simple analytical expressions are obtained for any d≥2 and n≥2 for the endpoint distributions of two families of walks whose q are integers or half-integers which depend solely on d. These endpoint distributions have a simple geometrical interpretation. Expressed for a two-step planar walk whose q=1, it means that the distribution of the endpoint on a disc of radius 1 is identical to the distribution of the projection on the disc of a point M uniformly distributed over the surface of the 3D unit sphere. Five additional walks, with a uniform distribution of the endpoint in the inside of a ball, are found from known finite integrals of products of powers and Bessel functions of the first kind. They include four different walks in ℝ3, two of two steps and two of three steps, and one walk of two steps in ℝ4. Pearson–Liouville random walks, obtained by distributing the total lengths of the previous Pearson–Dirichlet walks according to some specified probability law are finally discussed. Examples of unconstrained random walks, whose step lengths are gamma distributed, are more particularly considered.

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References

  1. Pearson, K.: The problem of the random walk. Nature 72, 294 (1905)

    Article  ADS  Google Scholar 

  2. Magnello, M.E.: Karl Pearson and the establishment of mathematical statistics. Int. Stat. Rev. 77, 3–29 (2009)

    Article  Google Scholar 

  3. Kiefer, J.E., Weiss, G.H.: The Pearson random walk. AIP Conf. Proc. 109, 11–32 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  4. Codling, E.A.: Biased random walks in biology. Ph.D., The University of Leeds (2003)

  5. Codling, E.A., Plank, M.J., Benhamou, S.: Random walk models in biology. J. R. Soc. Interface 5, 813–834 (2008)

    Article  Google Scholar 

  6. Bartumeus, F., Catalan, J., Viswanathan, G.M., Raposo, E.P., da Luz, M.G.E.: The influence of turning angles on the success of non-oriented animal searches. J. Theor. Biol. 252, 43–55 (2008)

    Article  Google Scholar 

  7. Byers, J.A.: Correlated random walk equations of animal dispersal resolved by simulation. Ecology 82, 1680–1690 (2001)

    Article  Google Scholar 

  8. Serino, C.A., Redner, S.: Pearson walk with shrinking steps in two dimensions. J. Stat. Mech. P01006 (2010)

  9. Stadje, W.: The exact probability distribution of a two-dimensional random walk. J. Stat. Phys. 46, 207–216 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  10. Franceschetti, M.: When a random walk of fixed length can lead uniformly anywhere inside a hypersphere. J. Stat. Phys. 127, 813–823 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. García-Pelayo, R.: Twice scattered particles in a plane are uniformly distributed. J. Stat. Phys. 127, 401–404 (2008)

    Article  ADS  Google Scholar 

  12. Orsingher, E., De Gregorio, A.: Random flights in higher spaces. J. Theor. Probab. 20, 769–806 (2007)

    Article  MATH  Google Scholar 

  13. Kolesnik, A.D.: Random motion at finite speed in higher dimensions. J. Stat. Phys. 131, 1039–1065 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Lefebvre, M.: Processus stochastiques appliqués. Hermann Editeurs, Paris (2005)

    Google Scholar 

  15. David, H.A., Nagaraja, H.N.: Order statistics, 3rd edn. Wiley, New York (2003)

    MATH  Google Scholar 

  16. Huillet, T., Paroissin, C.: Sampling from Dirichlet partitions: estimating the number of species. Environmetrics (2009). doi:10.1002/env.977

    Google Scholar 

  17. Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London (1986)

    MATH  Google Scholar 

  18. Letac, G.: Donkey walk and Dirichlet distributions. Stat. Probab. Lett. 57, 17–22 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Fang, K.-T., Kotz, S., Ng, K.-W.: Symmetric Multivariate and Related Distributions. Chapman and Hall, London (1990)

    MATH  Google Scholar 

  20. Johnson, N.L., Kotz, S.: Distributions in Statistics Continuous Univariate Distributions-1. Wiley, New York (1970)

    Google Scholar 

  21. Reed, W.J.: Random points in a simplex. Pac. J. Math. 54, 183–198 (1974)

    MATH  Google Scholar 

  22. James, I.R., Mosimann, J.E.: A new characterization of the Dirichlet distribution through neutrality. Ann. Stat. 8, 83–189 (1980)

    Article  MathSciNet  Google Scholar 

  23. Chang, W.Y., Gupta, R.D., Richards, D.St.P.: Structural properties of the generalized Dirichlet distributions. In: Viana, M.A.G., Wynn, H.P. (eds.) Algebraic Methods in Statistics and Probability II. Contemp. Math., vol. 516. American Mathematical Society, Providence (2010). ISBN-10: 0-8218-4891-7

    Google Scholar 

  24. Beghin, L., Orsingher, E.: Moving randomly amid scattered obstacles. Stochastics 82, 201–229 (2010)

    Google Scholar 

  25. Kolesnik, A.D.: The explicit probability distribution of a six-dimensional random flight. Theory Stoch. Proc. 15, 33–39 (2009)

    MathSciNet  Google Scholar 

  26. Watson, G.S.: Statistics on Spheres. Wiley, New York (1983)

    MATH  Google Scholar 

  27. Lord, R.D.: The distribution of distance in a hypersphere. Ann. Math. Stat. 25, 794–798 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  28. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1980)

    Google Scholar 

  29. Watson, G.N.: A treatise on the theory of Bessel functions. Cambride University Press, Cambridge (1995)

    MATH  Google Scholar 

  30. Diaconis, P., Freedman, D.: A dozen de Finetti-style results in search for a theory. Ann. Inst. Henri Poincaré 23, 397–423 (1987)

    MathSciNet  Google Scholar 

  31. Gupta, R.D., Richards, D.St.P.: Multivariate Liouville distributions. J. Multivar. Anal. 23, 233–256 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lord, R.D.: The use of the Hankel transform in statistics: I. General theory and examples. Biometrika 41, 344–350 (1954)

    MATH  MathSciNet  Google Scholar 

  33. Gneiting, T.: Normal scale mixtures and dual probability densities. J. Stat. Comput. Simul. 59, 375–384 (1997)

    Article  MATH  Google Scholar 

  34. Dreier, I., Kotz, S.: A note on the characteristic function of the t-distribution. Stat. Probab. Lett. 57, 221–224 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Huillet, T.: Statistical aspects of random fragmentations. J. Comput. Appl. Math. 181, 364–387 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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  1. Institut de Physique de Rennes, UMR UR1-CNRS 6251, Université de Rennes I, Campus de Beaulieu, Bâtiment 11A, 35042, Rennes Cedex, France

    Gérard Le Caër

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  1. Gérard Le Caër
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Le Caër, G. A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths. J Stat Phys 140, 728–751 (2010). https://doi.org/10.1007/s10955-010-0015-8

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