Tightness of measures

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

Contents

Definitions

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.

Examples

Compact spaces

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 spaces

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 masses

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 measures

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 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 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

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