The Benford phenomenon for random variables. Discussion of Feller's way

This is a detailed survey which mainly presents the Pinkham-Feller way. I added some new points to the first version [V2] and I suppressed "Examples" devoted to Gamma, Fr\'echet and Weibull laws. Theorem 2 is a bit more general (no assumption of density: this answers a question of T. Hill). Section 10 is new and devoted to an argument (Poincar\'e, Fewster) about the effect of high frequencies oscillations. Maybe many works, many efforts, have been devoted to the study of a sufficient condition of poor value: see Sections 7 and 11. The final Section gives some suggestions.Comment: 23 page

Le futur de l'autonomie de l'individu. Politique et nihilisme. Lecture philosophique et théologique

The future of the autonomy of the individual. Politics and nihilism. Philosophical and theological vision ABSTRACT: In this article, Paul Valadier looks into the logics and foundations of individualism and examines, at the same time, the consequences on postmodern society. He proposes the possibilities of a new concept of the autonomy of the individual by linking modern tradition and Catholic theology. KEY-WORDS: authonomy, nihilism, individualism Paul Valadier investiga, en aquest article, la lògica i els fonaments de l’individualisme i explora, a la vegada, les conseqüències que té en la societat postmoderna. Proposa les plossibilitats d’una nova concepció de l’autonomia de l’individu enllaçant la tradició modern i la teologia catòlica. PARAULES CLAU: Autonomia, nihilisme, individualism

No arbitrage and closure results for trading cones with transaction costs

In this paper, we consider trading with proportional transaction costs as in Schachermayer’s paper (Schachermayer in Math. Finance 14:19–48, 2004). We give a necessary and sufficient condition for ${\mathcal{A}}$ , the cone of claims attainable from zero endowment, to be closed. Then we show how to define a revised set of trading prices in such a way that, firstly, the corresponding cone of claims attainable for zero endowment, ${\tilde{ {\mathcal{A}}}}$ , does obey the fundamental theorem of asset pricing and, secondly, if ${\tilde{ {\mathcal{A}}}}$ is arbitrage-free then it is the closure of ${\mathcal{A}}$ . We then conclude by showing how to represent claims

Quantiles as minimizers

A real random variable admits median(s) and quantiles. These values minimize convex functions on $\mathbb R$. We show by "Convex Analysis" arguments that the function to be minimized is very natural. The relationship with some notions about functions of bounded variation developed by J.J.~Moreau is emphasized