With the outbreak of COVID-19 across Europe, anonymized telecommunications data provides a key insight into population level mobility and assessing the impact and effectiveness of containment measures. Vodafone’s response across its global footprint was fast and delivered key new metrics for the pandemic that have proven to be useful for a number of external entities. Cooperation with national governments and supra-national entities to help fight the COVID-19 pandemic was a key part of Vodafone’s response, and in this article the different methodologies developed are analyzed, as well as the key collaborations established in this context. In this article we also analyze the regulatory challenges found, and how these can pose a risk of the full benefits of these insights not being harnessed, despite clear and efficient Privacy and Ethics assessments to ensure individual safety and data privacy.

We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U[0, 1]d random particle. We show that the limiting configuration contains N − 1 coincident particles at a random location ξN ∈ [0, 1]d. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN, showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.

During the last few years, several variants of P. Lévy's formula for the stochastic area of complex Brownian motion have been obtained. These are of interest in various domains of applied probability, particularly in relation to polymer studies. The method used by most authors is the diagonalization procedure of Paul Lévy.

Here we derive one such variant of Lévy's formula, due to Chan, Dean, Jansons and Rogers, via a change of probability method, which reduces the computation of Laplace transforms of Brownian quadratic functionals to the computations of the means and variances of some adequate Gaussian variables.

We then show that with the help of linear algebra and invariance properties of the distribution of Brownian motion, we are able to derive simply three other variants of Lévy's formula.