Editing Concentration of measure (section)

{{Use American English|date=January 2019}}
{{Use mdy dates|date=January 2019}}
{{Short description|Statistical parameter}}
In [[mathematics]], '''concentration of measure''' (about a [[median]]) is a principle that is applied in [[measure theory]], [[probability]] and [[combinatorics]], and has consequences for other fields such as [[Banach space]] theory. Informally, it states that "A random variable that depends in a [[Lipschitz continuity|Lipschitz]] way on many independent variables (but not too much on any of them) is essentially constant".<ref>{{cite journal
| first1=Michel | last1=Talagrand | authorlink1=Michel Talagrand
| title=A New Look at Independence
| journal=[[Annals of Probability]]
| year=1996
| volume=24
| issue=1
| pages=1–34
| doi=10.1214/aop/1042644705 | doi-access=free}}</ref>

The concentration of measure phenomenon was put forth in the early 1970s by [[Vitali Milman]] in his works on the local theory of [[Banach space]]s, extending an idea going back to the work of [[Paul Lévy (mathematician)|Paul Lévy]].<ref>"''The concentration of <math>f_\ast(\mu)</math>, ubiquitous in the probability theory and statistical mechanics, was brought to geometry (starting from Banach spaces) by Vitali Milman, following the earlier work by Paul Lévy''" - [[Mikhail Gromov (mathematician)|M. Gromov]], Spaces and questions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part I, 118&ndash;161.</ref><ref>"''The idea of concentration of measure (which was discovered by V.Milman) is arguably one of the great ideas of analysis in our times. While its impact on Probability is only a small part of the whole picture, this impact should not be ignored.''" - [[Michel Talagrand|M. Talagrand]], A new look at independence, Ann. Probab. 24 (1996), no. 1, 1&ndash;34.</ref> It was further developed in the works of Milman and [[Mikhail Gromov (mathematician)|Gromov]], [[Bernard Maurey|Maurey]], [[Gilles Pisier|Pisier]], [[Gideon Schechtman|Schechtman]], [[Michel Talagrand|Talagrand]], [[Michel Ledoux|Ledoux]], and others.