Exact probability density function for a 3D random walk with geometrically shrinking steps with shrinking factor 1/2

Abstract

This paper presents the exact expressions for the probability density function of a 3D random walk with step lengths ℓ =1, λ, λ₂, . . . in taking the N = 1, 2, 3, . . . steps for a shrinking factor λ = 1/2. For any number of steps, the regions where the random walker may be located is given. Because the step lengths of the random walk is geometrically shrinking the walker is confined within a a certain region even for N → ∞. The random walk is compared with its 1D and 2D counterparts.