The Distributional Asymptotics Mod 1 of (logb n)

This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log b. An improved upper estimate (\sqrt {\log N} /N) is obtained for the rate of convergence with respect to (w. r. t.)the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N)w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.

Geodesics of minimal length in the set of probability measures on graphs

We endow the set of probability measures on a weighted graph with a Monge–Kantorovich metric induced by a function defined on the set of edges. The graph is assumed to have n vertices and so the boundary of the probability simplex is an affine (n − 2)-chain. Characterizing the geodesics of minimal length which may intersect the boundary is a challenge we overcome even when the endpoints of the geodesics do not share the same connected components. It is our hope that this work will be a preamble to the theory of mean field games on graphs.

An approximation scheme for the Kantorovich-Rubinstein problem on compact spaces

This paper presents an approximation scheme for the Kantorovich-Rubinstein mass transshipment (KR) problem on compact spaces. A sequence of finite-dimensional linear programs, minimal cost network flow problems with bounds, are introduced and it is proven that the limit of the sequence of the optimal values of these problems is the optimal value of the KR problem. Numerical results are presented approximating the Kantorovich metric between distributions on [0,1].