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Two-step Dirichlet random walks

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  • Le Caër, Gérard

Abstract

Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d⩾2), which are characterized by a sum of step lengths which is fixed and taken to be 1 without loss of generality, are named “Dirichlet” when this constraint is realized via a Dirichlet law of step lengths. The latter continuous multivariate distribution, which depends on n positive parameters, generalizes the beta distribution (n=2). It is simply obtained from n independent gamma random variables with identical scale factors. Previous literature studies of these random walks dealt with symmetric Dirichlet distributions whose parameters are all equal to a value q which takes half-integer or integer values. In the present work, the probability density function of the distance from the endpoint to the origin is first made explicit for a symmetric Dirichlet random walk of two steps. It is valid for any positive value of q and for all d⩾2. The latter pdf is used in turn to express the related density of a random walk of two steps whose step length is distributed according to an asymmetric beta distribution which depends on two parameters, namely q and q+s where s is a positive integer.

Suggested Citation

  • Le Caër, Gérard, 2015. "Two-step Dirichlet random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 430(C), pages 201-215.
    • Handle: RePEc:eee:phsmap:v:430:y:2015:i:c:p:201-215
    DOI: 10.1016/j.physa.2015.02.075
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    References listed on IDEAS

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    1. T. Pham-Gia & N. Turkkan, 2002. "The product and quotient of general beta distributions," Statistical Papers, Springer, vol. 43(4), pages 537-550, October.
    2. Pogorui, Anatoliy A. & Rodríguez-Dagnino, Ramón M., 2013. "Random motion with gamma steps in higher dimensions," Statistics & Probability Letters, Elsevier, vol. 83(7), pages 1638-1643.
    3. De Gregorio, Alessandro & Orsingher, Enzo, 2012. "Flying randomly in Rd with Dirichlet displacements," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 676-713.
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