Tightness of measures

In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity".

DefinitionsEdit

Let   be a Hausdorff space, and let   be a σ-algebra on   that contains the topology  . (Thus, every open subset of   is a measurable set and   is at least as fine as the Borel σ-algebra on  .) Let   be a collection of (possibly signed or complex) measures defined on  . The collection   is called tight (or sometimes uniformly tight) if, for any  , there is a compact subset   of   such that, for all measures  ,

 

where   is the total variation measure of  . Very often, the measures in question are probability measures, so the last part can be written as

 

If a tight collection   consists of a single measure  , then (depending upon the author)   may either be said to be a tight measure or to be an inner regular measure.

If   is an  -valued random variable whose probability distribution on   is a tight measure then   is said to be a separable random variable or a Radon random variable.

ExamplesEdit

Compact spacesEdit

If   is a metrisable compact space, then every collection of (possibly complex) measures on   is tight. This is not necessarily so for non-metrisable compact spaces. If we take   with its order topology, then there exists a measure   on it that is not inner regular. Therefore, the singleton   is not tight.

Polish spacesEdit

If   is a compact Polish space, then every probability measure on   is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on   is tight if and only if it is precompact in the topology of weak convergence.

A collection of point massesEdit

Consider the real line   with its usual Borel topology. Let   denote the Dirac measure, a unit mass at the point   in  . The collection

 

is not tight, since the compact subsets of   are precisely the closed and bounded subsets, and any such set, since it is bounded, has  -measure zero for large enough  . On the other hand, the collection

 

is tight: the compact interval   will work as   for any  . In general, a collection of Dirac delta measures on   is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measuresEdit

Consider  -dimensional Euclidean space   with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

 

where the measure   has expected value (mean)   and covariance matrix  . Then the collection   is tight if, and only if, the collections   and   are both bounded.

Tightness and convergenceEdit

Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimension. See

Exponential tightnessEdit

A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures   on a Hausdorff topological space   is said to be exponentially tight if, for any  , there is a compact subset   of   such that

 

ReferencesEdit

  • Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
  • Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.
  • Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015 (See chapter 2)