SEMINARIO
Speaker: Yinon Spinka
Title: Finitely-dependent processes are finitary
Venerdi' 18 Gennaio 2019 ORE 11:30
Dipartimento di Matematica e Fisica
Universita' degli Studi Roma Tre
Largo San Leonardo Murialdo,1 - Pal.C Aula 211
Abstract:
Consider a translation-invariant process $X$ indexed by $mathbb{Z}^d$.
Suppose that $X$ is finitely-dependent in the sense that its
restrictions to sets which are sufficiently separated (at least some
fixed distance apart) are independent. We are concerned with the
following question: How "close" is $X$ to being an i.i.d. process? One
natural notion of closeness, called block factor, was suggested by
Ibragimov and Linnik over 50 years ago. It took roughly 30 years until
Burton, Goulet and Meester constructed an example which showed that
this notion is too strong. That is, $X$ may not be close to being
i.i.d. in this sense. We show that $X$ is close in a slightly weaker
sense -- it is a finitary factor of an i.i.d. process. This means that
$X=F(Y)$ for some i.i.d. process $Y$ and some measurable map F which
commutes with translations of $mathbb{Z}^d$, and moreover, that in
order to determine the value of $X_v$ for a given $v$, one only needs
to look at a finite (but random) region of $Y$. The result extends to
finitely-dependent processes indexed by the vertex set of any
transitive amenable graph.