Tightness of measures explained

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 Hausdorff space, and let \Sigma be a σ-algebra on X that contains the topology T . (Thus, every open subset of X is a measurable set and \Sigma 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 \Sigma . The collection M is called tight (or sometimes uniformly tight) if, for any \varepsilon > 0 , there is a compact subset K _{\varepsilon} of X such that, for all measures \mu \in M ,|\mu| (X \setminus K _{\varepsilon} ) < \varepsilon.

where

|\mu| is the total variation measure of \mu . Very often, the measures in question are probability measures, so the last part can be written as\mu (K _{\varepsilon} ) > 1 - \varepsilon.

If a tight collection

M consists of a single measure \mu , then (depending upon the author) \mu 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.

Another equivalent criterion of the tightness of a collection

M is sequentially weakly compact. We say the family M of probability measures is sequentially weakly compact if for every sequence \left\{\mu _{n\right\}} from the family, there is a subsequence of measures that converges weakly to some probability measure \mu . It can be shown that a family of measure is tight if and only if it is sequentially weakly compact.

Examples

Compact spaces

If

X is a metrizable 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 [ 0 ,\omega _{1]} with its order topology, then there exists a measure \mu on it that is not inner regular. Therefore, the singleton \{\mu\} 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 ifit is precompact in the topology of weak convergence.

A collection of point masses

R with its usual Borel topology. Let \delta _{x} denote the Dirac measure, a unit mass at the point x in R . The collectionM _{1} : = \{ \delta _{n} | n \in N \}

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 \delta _{n} -measure zero for large enough n . On the other hand, the collectionM _{2} : = \{ \delta _{1} | n \in N \}

is tight: the compact interval

[ 0 , 1 ] will work as K _{\varepsilon} for any \varepsilon > 0 . In general, a collection of Dirac delta measures on R ^{n} is tight if, and only if, the collection of their supports is bounded.

A collection of Gaussian measures

Consider

n -dimensional Euclidean space R ^{n} with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures\Gamma = \{ \gamma _{i} | i \in I \},

where the measure

\gamma _{i} has expected value (mean) m _{i} \in R ^{n} and covariance matrix C _{i} \in R ^{n} . Then the collection \Gamma is tight if, and only if, the collections \{ m _{i} | i \in I \} \subseteq R ^{n} and \{ C _{i} | i \in I \} \subseteq R ^{n} 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

Exponential tightness

A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures

(\mu _{\delta} ) _{\delta} on a Hausdorff topological space X is said to be exponentially tight if, for any \varepsilon > 0 , there is a compact subset K _{\varepsilon} of X such that\limsup _{\delta} \delta log \mu _{\delta} (X \setminus K _{\varepsilon} ) < - \varepsilon.

References

• Book: Billingsley, Patrick . Probability and Measure . John Wiley & Sons, Inc. . New York, NY . 1995 . 0-471-00710-2.
• Book: Billingsley, Patrick . Convergence of Probability Measures . registration . John Wiley & Sons, Inc. . New York, NY . 1999 . 0-471-19745-9.
• Book: Ledoux. Michel. Talagrand . Michel . Michel Talagrand. Probability in Banach spaces. Springer-Verlag. Berlin. 1991. xii+480. 3-540-52013-9. (See chapter 2)