Tightness of measures
Let be a topological 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 ,
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 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.
If is a 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 masses
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 measures
Tightness and convergence
- Finite-dimensional distribution
- Prokhorov's theorem
- Lévy–Prokhorov metric
- weak convergence of measures
- Tightness in classical Wiener space
- Tightness in Skorokhod space
A generalization 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
- 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. MR 1102015 (See chapter 2)