Normal-inverse Gaussian distribution: Difference between revisions

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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]]."<!-- and beyond!-->
 
==Definitions==
 
Let (''X'', ''T'') be a [[topological space]], and let Σ be a [[sigma algebra|&sigma;-algebra]] on ''X'' that contains the topology ''T''. (Thus, every [[open set|open subset]] of ''X'' is a [[measurable set]] and Σ is at least as fine as the [[Borel sigma algebra|Borel &sigma;-algebra]] on ''X''.) Let ''M'' be a collection of (possibly [[signed measure|signed]] or [[complex measure|complex]]) measures defined on Σ. The collection ''M'' is called '''tight''' (or sometimes '''uniformly tight''') if, for any ''ε''&nbsp;&gt;&nbsp;0, there is a [[compact space|compact subset]] ''K''<sub>''ε''</sub> of ''X'' such that, for all measures ''μ'' in ''M'',
 
:<math>|\mu| (X \setminus K_{\varepsilon}) < \varepsilon.</math>
 
where <math>|\mu|</math> is the [[total variation measure]] of <math>\mu</math>. Very often, the measures in question are [[probability measure]]s, so the last part can be written as
 
:<math>\mu (K_{\varepsilon}) > 1 - \varepsilon. \,</math>
 
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 <math>[0,\omega_1]</math> with its [[order topology]], then there exists a measure <math>\mu</math> on it that is not inner regular. Therefore the singleton <math>\{\mu\}</math> 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 [[Relatively compact subspace|precompact]] in the topology of [[Convergence of measures|weak convergence]].
 
===A collection of point masses===
 
Consider the [[real line]] '''R''' with its usual Borel topology. Let ''δ''<sub>''x''</sub> denote the [[Dirac measure]], a unit mass at the point ''x'' in '''R'''. The collection
 
:<math>M_{1} := \{ \delta_{n} | n \in \mathbb{N} \}</math>
 
is not tight, since the compact subsets of '''R''' are precisely the [[Closed set|closed]] and [[Bounded set|bounded]] subsets, and any such set, since it is bounded, has ''δ''<sub>''n''</sub>-measure zero for large enough ''n''. On the other hand, the collection
 
:<math>M_{2} := \{ \delta_{1 / n} | n \in \mathbb{N} \}</math>
 
is tight: the compact interval [0, 1] will work as ''K''<sub>''η''</sub> for any ''η'' &gt; 0. In general, a collection of Dirac delta measures on '''R'''<sup>''n''</sup> is tight if, and only if, the collection of their [[support (measure theory)|supports]] is bounded.
 
===A collection of Gaussian measures===
 
Consider ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> with its usual Borel topology and σ-algebra. Consider a collection of [[Gaussian measure]]s
 
:<math>\Gamma = \{ \gamma_{i} | i \in I \},</math>
 
where the measure ''γ''<sub>''i''</sub> has [[expected value]] ([[mean]]) ''μ''<sub>''i''</sub> in '''R'''<sup>''n''</sup> and [[variance]] ''σ''<sub>''i''</sub><sup>2</sup> &gt; 0. Then the collection Γ is tight if, and only if, the collections <math>\{ \mu_{i} | i \in I \} \subseteq \mathbb{R}^{n}</math> and <math>\{ \sigma_{i}^{2} | i \in I \} \subseteq \mathbb{R}</math> are both bounded.
 
==Tightness and convergence==
 
Tightness is often a necessary criterion for proving the [[weak convergence of measures|weak convergence]] of a sequence of probability measures, especially when the measure space has [[Infinity|infinite]] [[dimension]]. See
 
* [[Finite-dimensional distribution]]
* [[Prokhorov's theorem]]
* [[Classical Wiener space#Tightness in classical Wiener space|Tightness in classical Wiener space]]
* [[Càdlàg#Tightness in Skorokhod space|Tightness in Skorokhod space]]
 
==Exponential tightness==
 
A generalization of tightness is the concept of exponential tightness, which has applications in [[large deviations theory]]. A family of [[probability measure]]s (''μ''<sub>''δ''</sub>)<sub>''δ''&gt;0</sub> on a [[Hausdorff space|Hausdorff]] topological space ''X'' is said to be '''exponentially tight''' if, for any ''η''&nbsp;&gt;&nbsp;0, there is a compact subset ''K''<sub>''η''</sub> of ''X'' such that
 
:<math>\limsup_{\delta \downarrow 0} \delta \log \mu_{\delta} (X \setminus K_{\eta}) < - \eta.</math>
 
==References==
 
* {{cite book | last=Billingsley | first=Patrick | title=Probability and Measure | publisher=John Wiley & Sons, Inc. | location=New York, NY | year=1995 | isbn=0-471-00710-2}}
* {{cite book | last=Billingsley | first=Patrick | title=Convergence of Probability Measures | publisher=John Wiley & Sons, Inc. | location=New York, NY | year=1999 | isbn=0-471-19745-9}}
* {{ cite book
| last1 = Ledoux
| first1 = Michel
| last2 = Talagrand | first2 = Michel | author2-link = Michel Talagrand
| title = Probability in Banach spaces
| publisher = Springer-Verlag
| location = Berlin
| year = 1991
| pages = xii+480
| isbn = 3-540-52013-9
}} {{MathSciNet|id=1102015}} (See chapter 2)
 
[[Category:Measure theory]]

Revision as of 05:05, 10 December 2013

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,

|μ|(X∖Kε)<ε.

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

μ(Kε)>1−ε.

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 [0,ω1] with its order topology, then there exists a measure μ on it that is not inner regular. Therefore the singleton {μ} 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

M1:={δn|n∈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 δn-measure zero for large enough n. On the other hand, the collection

M2:={δ1/n|n∈ℕ}

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

Γ={γi|i∈I},

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 {μi|i∈I}⊆ℝn and {σi2|i∈I}⊆ℝ 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

lim supδ↓0δlog⁡μδ(X∖Kη)<−η.

References

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