Year: 2026
Type: preprint
Abstract: In the Pearson random walk, the direction of the i-th step is a random variable uniformly distributed on the unit sphere of R^d, and its length is a non-negative random variable. We prove that, when d goes to infinity, the Euclidean norm of the walk converges almost surely to the Euclidean norm of the vector of step lengths, whose joint law is arbitrary. Further, we investigate the same convergence result in a more general framework, where both the number and the lengths of the steps depend on the dimension. The results in the present paper can be applied to approximate the distribution of some classical Pearson random walks, for example, in the Dirichlet case.
Source: arXiv (Cornell University)
Institution: University of Insubria
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211903417