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

be a Hausdorff space be a σ-algebra. (Thus, every open subset is a measurable set is at least as fine as the Borel σ-algebra be a collection of (possibly signed or complex is called tight (or sometimes uniformly tight, there is a compact subset,

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

may either be said to be a tight measure or to be an inner regular measure.

-valued random variable whose probability distribution is said to be a separable random variable or a Radon random variable.

. It can be shown that a family of measures is tight if and only if it is sequentially weakly compact.

Examples

Compact spaces

is a metrizable compact space with its order topology is not tight.

Polish spaces

is a Polish space is tight; this is Ulam's theorem. Furthermore, by Prokhorov's theorem is tight if and only if it is precompact in the topology of weak convergence.

A collection of point masses

Consider the real line denote the Dirac measure are precisely the closed and bounded is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

-dimensional Euclidean space with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures has expected value (mean and covariance matrix 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

denotes the stochastic order.

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 is said to be exponentially tight such that

References

• (See chapter 2)