Talagrand's concentration inequality
In probability theory, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first proved by the French mathematician Michel Talagrand. The inequality is one of the manifestations of the concentration of measure phenomenon.Statement
The inequality states that if is a product space endowed with a product probability measure and
is a subset in this space, then for any
where is the complement of where this is defined by
and where is Talagrand's convex distance defined as
where, are -dimensional vectors with entries
respectively and is the -norm. That is,
