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".
Definitions
Let be a Hausdorff space, and let be a σ-algebra on that contains the topology. Let be a collection of measures defined on. The collection is called 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 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.
Another equivalent criterion of the tightness of a collection is sequential weak compactness. We say the family of probability measures is sequentially weakly compact if for every sequence from the family, there is a subsequence of measures that converges weakly to some probability measure. It can be shown that a family of measures is tight if and only if it is sequentially weakly compact.
Examples
Compact spaces
If is a metrizable compact space, then every collection of 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 spaces
If is a Polish space, then every finite measure on is tight; this is Ulam's theorem. Furthermore, by Prokhorov's theorem, a collection of probability measures on is tight if and only ifit is precompact in the topology of weak convergence.
A collection of point masses
Consider the real line with its usual Borel topology. Let denote the Dirac measure, a unit mass at the point in. The collectionis 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 measures
Consider -dimensional Euclidean space with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measureswhere the measure has expected value and covariance matrix. Then the collection is tight if, and only if, the collections and are both bounded.
Tightness and convergence
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- Finite-dimensional distribution
- Prokhorov's theorem
- Lévy–Prokhorov metric
- Weak convergence of measures
- Tightness in classical Wiener space
- Tightness in Skorokhod space
Tightness and stochastic ordering
such that
for all, where
denotes the stochastic order defined by
if for all nondecreasing functions.