Tightness of measures

From formulasearchengine
Jump to navigation Jump to search

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 (X, T) be a topological space, and let Σ be a σ-algebra on X that contains the topology T. (Thus, every open subset of X is a measurable set and Σ is at least as fine as the Borel σ-algebra on X.) Let M be a collection of (possibly signed or complex) measures defined on Σ. The collection M is called tight (or sometimes uniformly tight) if, for any ε > 0, there is a compact subset Kε of X such that, for all measures μ in M,

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

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

Examples

Compact spaces

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

A collection of point masses

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

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

is tight: the compact interval [0, 1] will work as Kη for any η > 0. In general, a collection of Dirac delta measures on Rn is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

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

where the measure γi has expected value (mean) μi in Rn and variance σi2 > 0. Then the collection Γ 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

Exponential tightness

A generalization of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures (μδ)δ>0 on a Hausdorff topological space X is said to be exponentially tight if, for any η > 0, there is a compact subset Kη of X such that

References

  • {{#invoke:citation/CS1|citation

|CitationClass=book }}

  • {{#invoke:citation/CS1|citation

|CitationClass=book }}

  • {{#invoke:citation/CS1|citation

|CitationClass=book }} Template:MathSciNet (See chapter 2)