Distribution constant - Revision history

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en>SzMithrandir at 23:14, 8 February 2014

2014-02-08T23:14:51Z

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	~~In [[probability theory]],~~ '~~''Donsker~~'s ~~theorem''', named after [[Monroe D. Donsker]], identifies~~ a ~~certain [[stochastic process]] as a limit of [[empirical process]]es~~. It is ~~sometimes called~~ the ~~'''functional central limit theorem'''~~.		The title of the author is Figures but it's not the most masucline title out there. It's not a home std test typical factor but what she likes performing is base jumping and now she is trying to earn cash with it. Managing individuals is what I do and the salary has been truly fulfilling. home at home std testing [http://www.rxlist.com/collection-of-images/genital_warts_hpv_picture/pictures.htm std test] kit Puerto Rico is exactly [http://www.associazioneitalianafotografi.it/community/gruppi/points-to-know-when-confronted-with-candidiasis/ std home test] where he and his wife live.<br><br>Here [http://www.smylestream.org/groups/solid-advice-in-relation-to-candida/ http://www.smylestream.org/groups/solid-advice-in-relation-to-candida/] is my blog post - at home std testing ([http://mirim.ir/index.php?do=/profile-699/info/ check out this site])

	~~A centered and scaled version of~~ [~~[empirical distribution function]] ''F''<sub>''n''<~~/~~sub> defines an [[empirical process]]~~

	~~: <math> G_n(x)= \sqrt n ( F_n(x) - F(x) ) \, <~~/~~math>~~

	~~indexed by ''x'' ∈ '''R'''~~.

	~~'''Theorem''' (Donsker, Skorokhod, Kolmogorov) The sequence of ''G''<sub>''n''<~~/~~sub>(''x''), as random elements~~ of ~~the [[Skorokhod space]] <math>\mathcal{D}(~~-~~\infty,\infty)<~~/~~math>, [[convergence in distribution\|converges in distribution]] to a [[Gaussian process]] ''G'' with zero mean and covariance given by~~
	~~: <math>\operatorname{cov}[G(s), G(t)] = E[G(s) G(t)] = \min\{F(s), F(t)\} - F(s)F(t). \,<~~/~~math>~~
	~~The process ''G''(''x'') can be written as ''B''(''F''(''x'')) where ''B'' is a standard [[Brownian bridge]] on the unit interval~~.

	~~==History==~~
	~~By the classical [[central limit theorem]], for fixed ''x'', the random variable ''G''<sub>''n''</sub>(''x'') [[converges in distribution]] to a [[normal distribution\|Gaussian (normal)]~~] ~~[[random variable]] ''G''(''x'') with zero mean and variance ''F''(''x'')(1 − ''F''(''x'')) as the sample size ''n'' grows.~~

	~~Kolmogorov (1933) showed that when ''F''~~ is [~~[continuous function\|continuous]], the supremum <math>\scriptstyle\sup_t G_n(t)<~~/~~math> and supremum of absolute value, <math>\scriptstyle\sup_t \|G_n(t)\|<~~/math> [[convergence in distribution\|converges in distribution]] to the laws of the same functionals of the [[Brownian bridge]] ''B''(''t''), see the [[Kolmogorov–Smirnov test]]. In 1949 Doob asked whether the convergence in distribution held for more general functionals, thus formulating a problem of [[weak convergence of measures\|weak convergence]] of random functions in a suitable [[function space]].~~<ref>{{cite journal~~
	~~\|first=Joseph L~~. ~~\|last=Doob\|authorlink=Joseph L. Doob~~
	~~\|title=Heuristic approach to the Kolmogorov–Smirnov theorems~~
	~~\|journal=[[Annals of Mathematical Statistics]]~~
	~~\|volume=20 \|issue= \|pages=393–403 \|year=1949~~
	~~\|doi=10.1214~~/~~aoms~~/~~1177729991 \|mr=30732 \| zbl = 0035.08901~~
	~~}}<~~/~~ref>~~

	~~In 1952 Donsker stated and proved (not quite correctly)<ref name="dudley1999">{{cite book~~
	~~\|first=R.M. \|last=Dudley\|authorlink=Richard M. Dudley~~
	~~\|title=Uniform Central Limit Theorems~~
	~~\|publisher=Cambridge University Press~~
	~~\|year=1999~~
	~~\|isbn=0~~-~~521~~-~~46102~~-2
	~~}}<~~/~~ref> a general extension for the Doob-Kolmogorov heuristic approach~~. ~~In the original paper, Donsker proved that the convergence in law of ''G~~<~~sub~~>n<~~/sub~~>~~'' to the Brownian bridge holds for [[uniform distribution (continuous)\|Uniform[0,1]]] distributions with respect to uniform convergence in ''t'' over the interval [0,1].<ref>{{cite journal~~
	~~\|first=M. D. \|last=Donsker \|authorlink=Monroe D. Donsker~~
	~~\|title=Justification and extension of Doob's heuristic approach to the Kolmogorov–Smirnov theorems~~
	~~\|journal=~~[~~[Annals of Mathematical Statistics]]~~
	~~\|volume=23 \|issue= \|pages=277–281 \|year=1952~~
	~~\|doi=10.1214~~/~~aoms~~/~~1177729445 \|mr=47288 \| zbl = 0046~~.~~35103~~
	~~}}<~~/~~ref>~~

	However Donsker's formulation was not quite correct because of the problem of measurability of the functionals of discontinuous processes. In 1956 Skorokhod and Kolmogorov defined a separable metric ''d'', called the ''Skorokhod metric'', on the space of [[cadlag function]]s on [0,1], such that convergence for ''d'' to ~~a continuous function is equivalent to convergence for the sup norm, and showed that ''G<sub>n<~~/~~sub>'' converges in law in <math>\mathcal{D}[0,1]<~~/~~math> to the Brownian bridge~~.

	~~Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric~~. ~~One can prove<ref name="dudley1999"~~ /~~> that there exist ''X<sub>i<~~/~~sub>'', iid uniform~~ in ~~[0,1] and a sequence of sample~~-~~continuous Brownian bridges ''B''<sub>''n''</sub>, such that~~
	~~:<math>\\|G_n~~-~~B_n\\|_\infty<~~/~~math>~~
	is ~~measurable and~~ [~~[convergence in probability\|converges in probability]] to 0~~. ~~An improved version of this result, providing more detail on the rate of convergence, is the [[Komlós–Major–Tusnády approximation]]~~.

	=~~=See also==~~
	*[[Glivenko–Cantelli theorem]]
	*[[Kolmogorov–Smirnov test]]

	~~== References ==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Donsker's Theorem}}~~
	~~[[Category:Probability theorems]]~~
	~~[[Category:Statistical theorems]]~~
	~~[[Category:Empirical process]~~]

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	~~Greetings! I am Marvella and I really feel comfortable when individuals use~~ the ~~full title~~. ~~South Dakota is exactly where I~~'~~ve always been residing~~. ~~One~~ of the ~~things she loves most is~~ to ~~study comics~~ and ~~she~~'ll be ~~beginning something else along~~ with it. ~~Managing people has been his day occupation~~ for a ~~whilst~~.<br><br>~~my webpage~~ - [~~http:~~//~~vip~~.~~akwacity~~.~~com~~/~~oxwall~~/~~blogs~~/~~post~~/~~10278 vip~~.~~akwacity~~.~~com~~]		In [[probability theory]], '''Donsker's theorem''', named after [[Monroe D. Donsker]], identifies a certain [[stochastic process]] as a limit of [[empirical process]]es. It is sometimes called the '''functional central limit theorem'''.

			A centered and scaled version of [[empirical distribution function]] ''F''<sub>''n''</sub> defines an [[empirical process]]

			: <math> G_n(x)= \sqrt n ( F_n(x) - F(x) ) \, </math>

			indexed by ''x'' ∈ '''R'''.

			'''Theorem''' (Donsker, Skorokhod, Kolmogorov) The sequence of ''G''<sub>''n''</sub>(''x''), as random elements of the [[Skorokhod space]] <math>\mathcal{D}(-\infty,\infty)</math>, [[convergence in distribution\|converges in distribution]] to a [[Gaussian process]] ''G'' with zero mean and covariance given by
			: <math>\operatorname{cov}[G(s), G(t)] = E[G(s) G(t)] = \min\{F(s), F(t)\} - F(s)F(t). \,</math>
			The process ''G''(''x'') can be written as ''B''(''F''(''x'')) where ''B'' is a standard [[Brownian bridge]] on the unit interval.

			==History==
			By the classical [[central limit theorem]], for fixed ''x'', the random variable ''G''<sub>''n''</sub>(''x'') [[converges in distribution]] to a [[normal distribution\|Gaussian (normal)]] [[random variable]] ''G''(''x'') with zero mean and variance ''F''(''x'')(1 − ''F''(''x'')) as the sample size ''n'' grows.

			Kolmogorov (1933) showed that when ''F'' is [[continuous function\|continuous]], the supremum <math>\scriptstyle\sup_t G_n(t)</math> and supremum of absolute value, <math>\scriptstyle\sup_t \|G_n(t)\|</math> [[convergence in distribution\|converges in distribution]] to the laws of the same functionals of the [[Brownian bridge]] ''B''(''t''), see the [[Kolmogorov–Smirnov test]]. In 1949 Doob asked whether the convergence in distribution held for more general functionals, thus formulating a problem of [[weak convergence of measures\|weak convergence]] of random functions in a suitable [[function space]].<ref>{{cite journal
			\|first=Joseph L. \|last=Doob\|authorlink=Joseph L. Doob
			\|title=Heuristic approach to the Kolmogorov–Smirnov theorems
			\|journal=[[Annals of Mathematical Statistics]]
			\|volume=20 \|issue= \|pages=393–403 \|year=1949
			\|doi=10.1214/aoms/1177729991 \|mr=30732 \| zbl = 0035.08901
			}}</ref>

			In 1952 Donsker stated and proved (not quite correctly)<ref name="dudley1999">{{cite book
			\|first=R.M. \|last=Dudley\|authorlink=Richard M. Dudley
			\|title=Uniform Central Limit Theorems
			\|publisher=Cambridge University Press
			\|year=1999
			\|isbn=0-521-46102-2
			}}</ref> a general extension for the Doob-Kolmogorov heuristic approach. In the original paper, Donsker proved that the convergence in law of ''G<sub>n</sub>'' to the Brownian bridge holds for [[uniform distribution (continuous)\|Uniform[0,1]]] distributions with respect to uniform convergence in ''t'' over the interval [0,1].<ref>{{cite journal
			\|first=M. D. \|last=Donsker \|authorlink=Monroe D. Donsker
			\|title=Justification and extension of Doob's heuristic approach to the Kolmogorov–Smirnov theorems
			\|journal=[[Annals of Mathematical Statistics]]
			\|volume=23 \|issue= \|pages=277–281 \|year=1952
			\|doi=10.1214/aoms/1177729445 \|mr=47288 \| zbl = 0046.35103
			}}</ref>

			However Donsker's formulation was not quite correct because of the problem of measurability of the functionals of discontinuous processes. In 1956 Skorokhod and Kolmogorov defined a separable metric ''d'', called the ''Skorokhod metric'', on the space of [[cadlag function]]s on [0,1], such that convergence for ''d'' to a continuous function is equivalent to convergence for the sup norm, and showed that ''G<sub>n</sub>'' converges in law in <math>\mathcal{D}[0,1]</math> to the Brownian bridge.

			Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove<ref name="dudley1999" /> that there exist ''X<sub>i</sub>'', iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges ''B''<sub>''n''</sub>, such that
			:<math>\\|G_n-B_n\\|_\infty</math>
			is measurable and [[convergence in probability\|converges in probability]] to 0. An improved version of this result, providing more detail on the rate of convergence, is the [[Komlós–Major–Tusnády approximation]].

			==See also==
			*[[Glivenko–Cantelli theorem]]
			*[[Kolmogorov–Smirnov test]]

			== References ==
			{{reflist}}

			{{DEFAULTSORT:Donsker's Theorem}}
			[[Category:Probability theorems]]
			[[Category:Statistical theorems]]
			[[Category:Empirical process]]

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Greetings! I am Marvella and I really feel comfortable when individuals use the full title. South Dakota is exactly where I've always been residing. One of the things she loves most is to study comics and she'll be beginning something else along with it. Managing people has been his day occupation for a whilst.<br><br>my webpage - [http://vip.akwacity.com/oxwall/blogs/post/10278 vip.akwacity.com]