217.111.196.30 at 09:00, 20 November 2014

2014-11-20T09:00:21Z

en>BeyondNormality: /* References */

2013-12-10T04:05:09Z

References

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	~~Golda~~ is ~~what~~'~~s written~~ on ~~my beginning certificate even though it~~ is ~~not~~ the ~~title~~ on ~~my birth certificate~~. It'~~s not~~ a ~~common thing but what she likes doing~~ is ~~to perform domino but she doesn~~'~~t have~~ the ~~time lately~~. ~~For years she'~~s ~~been operating~~ as a ~~journey agent~~. ~~For years she~~'~~s been living in Kentucky but her spouse wants them~~ to ~~transfer~~.<br><br>~~Here~~ is ~~my web blog~~ ... [~~http~~://~~myoceancounty~~.~~net~~/~~groups~~/~~apply-these-guidelines-when-gardening~~-and-~~grow~~/ ~~free psychic readings~~] ~~readings~~ [~~http:~~//~~www~~.~~january-yjm.com~~/xe/~~index~~.~~php?mid~~=~~video&document_srl~~=~~158289 online psychic reading~~] ([~~http~~:~~//findyourflirt~~.~~net~~/~~index~~.~~php?m~~=~~member_profile~~&p=~~profile&~~id=~~117823 findyourflirt.net~~])		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\|σ-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 σ-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'' > 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'' > 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> > 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''>0</sub> on a [[Hausdorff space\|Hausdorff]] topological space ''X'' is said to be '''exponentially tight''' if, for any ''η'' > 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]]

en>Much noise: Removed some junk.

2012-04-10T13:43:19Z

Removed some junk.

New page

Golda is what's written on my beginning certificate even though it is not the title on my birth certificate. It's not a common thing but what she likes doing is to perform domino but she doesn't have the time lately. For years she's been operating as a journey agent. For years she's been living in Kentucky but her spouse wants them to transfer.<br><br>Here is my web blog ... [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ free psychic readings] readings [http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289 online psychic reading] ([http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 findyourflirt.net])

Normal-inverse Gaussian distribution - Revision history

217.111.196.30 at 09:00, 20 November 2014

en>BeyondNormality: /* References */

en>Much noise: Removed some junk.