[Todos] Seminario de Probabilidad

[Matthieu Jonckheere]

PROXIMO ENCUENTRO: Miércoles 8 de Mayo, 12:00hs. (El 1 de mayo es feriado)

EXPOSITOR: Quinn Culver, University of  Notre Dame.

TITULO: Algorithmically Random Measures

LUGAR: Departamento de Matemática, Aula de seminarios, 2do piso, Pabellón 1.

RESUMEN: *

Algorithmic randomness attempts to answer the question "What exactly is a
random real number?" It does so by declaring that a real is nonrandom if it
can be captured in a null set that can be effectively (aka computably)
approximated from without by open sets. There is a rich interplay between
the theory of algorithmic randomness, the theory of computation (Turing
degrees), and probability theory. Recently, the theory of algorithmic
randomness with respect to other (i.e. non-uniform) measures has been
developed. This development, coupled with the fact that the space of
probability measures on the unit interval a nice enough space on which to
do computability, motivates the questions "Is there a natural measure on
the space of measures? What do its algorithmically random elements look
like?"
*
*
In attempt to answer these questions, we define a natural, computable map
that associates to each real a Borel probability measure, so that we can
talk about algorithmically random measures. We show that such random
measures are atomless and mutually singular with respect to Lebesgue. We
introduce a certain information-theoretic-ish property that lies strictly
between atomlessness and absolute continuity (with respect to the Lebesgue
measure) that we conjecture random measures satisfy. We then discuss other
maps, ask the question "Why is the first map more natural?", and discuss
some ways in which that question might be precisifiable.*
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