A closed form for the density functions of random walks in odd dimensions (Q2800103)

An \(n\)-step uniform random walk in \(\mathbb R^d\) starts at the origin and consists of \(n\) steps of unit length into a uniformly chosen random direction. The object of study is the corresponding continuous Lebesgue density of the distance to the origin. It is well known that in even dimension \(d\) the density typically involves elliptic integrals and in odd dimension \(d\) it can be represented in terms of elementary functions. The starting point of the present investigation is a convolution formula for the density in odd dimensions \(d\) involving the derivatives of a piecewise polynomial obtained by convolution. In a first step, the authors simplify the convolution using a recursion formula of its Fourier transform for which an explicit solution is given and checked by Maple code. The main result is an almost closed formula for the density only depending on the coefficients of a product of powers of a given polynomial and its reflection. To demonstrate the tractability of their formula, the authors compute explicit solutions for the density in dimensions \(d=3\) and \(d=5\) for at most \(n=4\) steps.