39,161 research outputs found
Comparing Fr\'echet and positive stable laws
Let be the unit exponential random variable and
the standard positive -stable random variable. We prove that
is decreasing for the optimal stochastic order and that is increasing for the convex
order, with We also show that
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 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 and and to some uniform estimates on the classical
Mittag-Leffler function. Along the way, we obtain a canonical factorization of
for 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 âasâ the other. Tim Ingoldâs (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 âpuzzle-solving⊠carried on within the context of involvement in a real world of persons, objects and relations.â (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 âunderstanding in practiceâ as a knowledge âbased on rich expectations generated over time about its shapeâ (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
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