Comparing Fr\'echet and positive stable laws

Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_\alpha$ the standard positive $\alpha$-stable random variable. We prove that $\{(1-\alpha) \alpha^{\gamma_\alpha} {\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha <1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha){\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha < 1\}$ is increasing for the convex order, with $\gamma_\alpha = \alpha/(1-\alpha).$ We also show that $\{\Gamma(1+\alpha) {\bf Z}_\alpha^{-\alpha}, 1/2\le \alpha \le 1\}$ is decreasing for the convex order, that {\bf Z}_\alpha^{-\alpha}\,\prec_{st}\, \Gamma(1-\alpha) \L and that \Gamma(1+\a){\bf Z}_\alpha^{-\alpha} \,\prec_{cx}\,{\bf L}. This allows to compare ${\bf Z}_\alpha$ with the two extremal Fr\'echet distributions corresponding to the behaviour of its density at zero and at infinity. We also discuss the applications of these bounds to the strange behaviour of the median of ${\bf Z}_\alpha$ and ${\bf Z}_\alpha^{-\alpha}$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\bf Z}_\alpha$ for $\alpha$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.Comment: To appear in Electronic Journal of Probabilit

The lower tail problem for homogeneous functionals of stable processes with no negative jumps

Let Z be a strictly a-stable real Levy process (a>1) and X be a fluctuating b-homogeneous additive functional of Z. We investigate the asymptotics of the first passage-time of X above 1, and give a general upper bound. When Z has no negative jumps, we prove that this bound is optimal and does not depend on the homogeneity parameter b. This extends a result of Y. Isozaki and solves partially a conjecture of Z. Shi.Comment: Revised version. To appear in ALEA Latin American Journal of Probability and Mathematical Statistic

Developing the formal structures of artistic practice-as-research

In this article I discuss a topic that is emerging as a valuable paradigm for creative practitioners - practice-as-research. There is some controversy over this term that, I believe, goes to the heart of our understanding of the nature of knowledge. The controversy relates to the idea that practice and research are two inherently different types of activity and therefore that it impossible to engage in one ‘as’ the other. Tim Ingold’s (2011) work on the anthropology of knowledge and skill alongside a broader stream of work on cognition and perception (see for example Lakoff & Johnson 2003 and Gibson 1979) suggests that both artistic practice and academic research involve ‘puzzle-solving… carried on within the context of involvement in a real world of persons, objects and relations.’ (Ingold 2011, p.419). The argument revolves around the notion that there is no such thing as disembodied or abstract knowledge and that all knowledge is both embodied and personally related to the world one inhabits. As such, the written word provides a schematic system for representing the much richer communication processes of speech and bodily experience. The written word, however, can only be understood through reference to our lived experience. Lave (1990, p.310) has termed this ‘understanding in practice’ as a knowledge ‘based on rich expectations generated over time about its shape’ (Lave 1990, p.323). Scholarly research outputs and their modes of publication are still firmly entrenched in the printed word. I will explore strategies for communicating the non-verbal knowledge that forms the basis of much practice-as-research