Measure concentration for Euclidean distance in the case of dependent random variables

By K. Marton. 2004

55 citations recorded by the source corpus.

Let qn be a continuous density function in n-dimensional Euclidean space. We think of qn as the density function of some random sequence Xn with values in $\Bbb{R}^{n}$. For I⊂[1,n], let XI denote the collection of coordinates Xi, i∈I, and let $\widebar X_{I}$ denote the collection of coordinates Xi, i∉I. We denote by $Q_{I}(x_{I}|\bar{x}_{I})$ the joint conditional density function of XI, given $\widebar X_{I}$. We prove measure concentration for qn in the case when, for an appropriate class of sets I, (i) the conditional densities $Q_{I}(x_{I}|\bar{x}_{I})$, as functions of xI, uniformly satisfy a logarithmic Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman’s strong mixing condition.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ebaadd0b8e59ff867bfd4122f492810e077c8e0c

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In Applied Mathematics

  1. Nathael Gozlan
  2. Christian Léonard
  3. L. Kontorovich
  4. Kavita Ramanan
  5. Arnaud Guillin
  6. Patrick Cattiaux
  7. Cyril Roberto
  8. Emmanuel Boissard

In Geometry and Topology

  1. Christian Léonard
  2. L. Kontorovich
  3. Kavita Ramanan
  4. Nathael Gozlan

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