2 research outputs found
A Pearson-Dirichlet random walk
A constrained diffusive random walk of n steps and a random flight in Rd,
which can be expressed in the same terms, were investigated independently in
recent papers. The n steps of the walk are identically and independently
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 are distributed over the unit (n-1) simplex according to a
Dirichlet distribution whose parameters are all equal to q, a given positive
value. The walk and the flight above correspond to q=1. For any d >= 3, there
exist, for integer and half-integer values of q, two families of
Pearson-Dirichlet walks which share a common property. For any n, the d
components of the endpoint are jointly distributed as are the d components of a
vector uniformly distributed over the surface of a hypersphere of radius l in a
space Rk whose dimension k is an affine function of n for a given d. 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 and two
walks in R4. Pearson-Liouville random walks, obtained by distributing the total
lengths of the previous Pearson-Dirichlet walks, are finally discussed.Comment: 33 pages 1 figure, the paper includes the content of a recently
submitted work together with additional results and an extended section on
Pearson-Liouville random walk
Karl Pearson Evolutionary biology and the emergence of a modern theory of statistics (1884-1936)
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