Citrabhanu - Revision history

en>David Eppstein: source

2014-12-13T01:08:21Z

source

← Older revision		Revision as of 03:08, 13 December 2014
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	In [[mathematics]], '''classical Wiener space''' is the collection of all [[continuous function]]s on a given [[domain of a function\|domain]] (usually a sub-[[interval (mathematics)\|interval]] of the [[real line]]), taking values in a [[metric space]] (usually ''n''-dimensional [[Euclidean space]]). Classical Wiener space is ~~useful in the study of [[stochastic processes]] whose sample paths are continuous functions. It is named after the [[United States\|American]] [[mathematician]] [[Norbert Wiener]].~~		Myrtle Benny is how I'm called and I feel comfortable when people use the full title. Years in the past we moved to North Dakota and I love every day living here. One of the extremely very best things in [http://www.1a-pornotube.com/blog/84958 over the counter std test] globe for him is to collect badges but he is struggling to find time for it. Managing people has been his working day job for a while.
	~~[[Image:Norbert wiener.jpg\|thumb\|upright\|Norbert Wiener]]~~

	~~==Definition==~~
	~~Given ''E'' ⊆ '''R'''<sup>''n~~'~~'</sup>~~ and ~~a metric space (''M'', ''d''),~~ the ~~'''classical Wiener space''' ''C''(''E''; ''M'') is the space of all continuous functions ''f'' : ''E'' → ''M'': i.e~~.~~, for every (fixed) ''t''~~ in ~~''E'',~~

	~~:<math>d(f(s), f(t)) \~~to ~~0</math> as <math>\| s - t \| \to 0.</math>~~

	~~In almost all applications, one takes ''E'' = [0, ''T''] or [0, +∞)~~ and ''M'' = '''R'''<sup>''n''</sup> for some ''n'' in '''N'''. For brevity, write ''C'' for ''C''([0, ''T'']; '''R'''<sup>''n''</sup>); this is a [[vector space]]. Write ''C''<sub>0</sub> for the [[linear subspace]] consisting only of those functions that take the value zero at the infimum of the set ''E''. Many authors refer to ''C''<sub>0</sub> as "classical Wiener space".

	~~==Properties~~ of ~~classical Wiener space==~~
	~~===Uniform topology===~~
	~~The vector space ''C'' can be equipped with~~ the ~~[[uniform norm]]~~

	~~:<math>\\| f \\| := \sup_{t \in [0, T]} \| f(t) \|</math>~~

	~~turning it into a [[normed vector space]] (~~in ~~fact a [[Banach space]]). This norm induces a~~ [~~[metric (mathematics)\|metric]] on ''C'' in the usual way~~: ~~<math>d (f, g) := \\| f-g \\|<~~/~~math>. The [[topology]] generated by the [[open set]]s in this metric is the topology of [[uniform convergence]] on [0, ''T''], or the [[uniform topology]].~~

	~~Thinking of the domain [0, ''T''] as "time" and the range '''R'''<sup>''n''<~~/sup> as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space a bit" and get the graph of ''f'' to lie on top of the graph of ''g'', while leaving time fixed. Contrast this with the [[Càdlàg#Skorokhod space\|Skorokhod topology]], which allows us to "wiggle" both space and time.

	~~===Separability and completeness===~~
	~~With respect to the uniform metric, ''C'' is both a [[separable space\|separable]] and a [[complete space]]:~~
	* separability is a consequence of the [[Stone-Weierstrass theorem]];
	* completeness is a consequence of the fact that the uniform limit of a sequence of continuous functions is itself continuous.

	~~Since it is both separable and complete, ''C'' is a [[Polish space]].~~

	~~===Tightness in classical Wiener space===~~
	~~Recall that the [[modulus of continuity]] for a function ''f'' : [0, ''T''] → '''R'''<sup>''n''</sup> is defined by~~

	~~:<math>\omega_{f} (\delta) := \sup \left\{ \| f(s) - f(t) \| \left\| s, t \in [0, T], \| s~~ - ~~t \| \leq \delta \right~~. ~~\right\}.<~~/~~math>~~

	~~This definition makes sense even if ''f'' is not continuous, and it can be shown that ''f'' is continuous [[if and only if]] its modulus of continuity tends to zero as δ → 0:~~

	~~:<math>f \in C \iff \omega_{f} (\delta) \to 0<~~/~~math> as δ → 0.~~

	~~By an application of~~ the ~~[[Arzelà-Ascoli theorem]], one can show that a sequence <math>(\mu_{n})_{n = 1}^{\infty}</math> of [[probability measure]]s on classical Wiener space ''C'' is [[tightness of measures\|tight~~]~~] if and only if both the following conditions are met:~~

	~~:<math>\lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n} \{ f \in C \| \| f(0) \| \geq a \} = 0,</math> and~~
	~~:<math>\lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n} \{ f \in C \| \omega_{f} (\delta) \geq \varepsilon \} = 0</math>~~ for ~~all ε > 0.~~

	~~===Classical Wiener measure===~~
	~~There~~ is ~~a "standard" measure on ''C''<sub>0</sub>, known as '''classical Wiener measure''' (or simply '''Wiener measure'''). Wiener measure has (at least) two equivalent characterizations:~~

	~~If one defines [[Brownian motion]]~~ to ~~be a [[Markov property\|Markov]] [[stochastic process]] ''B'' : [0, ''T''] × Ω → '''R'''<sup>''n''</sup>, starting at the origin, with [[almost surely]] continuous paths and independent increments~~

	~~:<math>B_{t} - B_{s} \sim \mathrm{Normal} \left( 0, \| t - s \| \right),</math>~~

	~~then classical Wiener measure γ is the [[law (stochastic processes)\|law]] of the process ''B''.~~

	~~Alternatively, one may use the [[abstract Wiener space]] construction, in which classical Wiener measure γ~~ is the [[Radonifying function\|radonification]] of the [[Cylinder set measure#Cylinder set measures on Hilbert spaces\|canonical Gaussian cylinder set measure]] on the Cameron-Martin [[Hilbert space]] corresponding to ~~''C''<sub>0</sub>.~~

	~~Classical Wiener measure is a [[Gaussian measure]]: in particular,~~ it ~~is a [[strictly positive measure\|strictly positive]] probability measure~~.

	~~Given classical Wiener measure γ on ''C''<sub>0</sub>, the [[product measure]] γ<sup>''n''</sup> × γ is~~ a ~~probability measure on ''C'', where γ<sup>''n''</sup> denotes the standard [[Gaussian measure]] on '''R'''<sup>''n''</sup>~~.

	~~==See also==~~
	* [[Càdlàg\|Skorokhod space]], a generalization of classical Wiener space, which allows functions to be discontinuous
	* [[Abstract Wiener space]]
	* [[Wiener process]]

	~~==References==~~
	* {{cite book \| author=Billingsley, Patrick \| title=Probability and Measure \| publisher=John Wiley & Sons, Inc., New York \| year=1995 \| isbn=0-471-00710-2}}

	~~[[Category:Stochastic processes]]~~
	~~[[Category:Metric geometry]]~~

en>Waacstats: add persondata short description using AWB

2013-07-13T22:51:10Z

add persondata short description using AWB

← Older revision		Revision as of 00:51, 14 July 2013
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	~~The author~~ is ~~known by~~ the ~~name~~ of ~~Figures Lint~~. ~~Playing baseball~~ is the ~~pastime he will~~ by ~~no means quit doing~~. ~~Bookkeeping~~ is ~~what I do~~. ~~Years ago he moved~~ to ~~North Dakota~~ and ~~his family enjoys~~ it.<br><br>~~Here~~ is ~~my site~~ - ~~home std test kit~~ ([~~https~~://~~www~~.~~epapyrus~~.~~com~~/xe/~~Purchase~~/~~5258960 hop over~~ to ~~these guys~~])		In [[mathematics]], '''classical Wiener space''' is the collection of all [[continuous function]]s on a given [[domain of a function\|domain]] (usually a sub-[[interval (mathematics)\|interval]] of the [[real line]]), taking values in a [[metric space]] (usually ''n''-dimensional [[Euclidean space]]). Classical Wiener space is useful in the study of [[stochastic processes]] whose sample paths are continuous functions. It is named after the [[United States\|American]] [[mathematician]] [[Norbert Wiener]].
			[[Image:Norbert wiener.jpg\|thumb\|upright\|Norbert Wiener]]

			==Definition==
			Given ''E'' ⊆ '''R'''<sup>''n''</sup> and a metric space (''M'', ''d''), the '''classical Wiener space''' ''C''(''E''; ''M'') is the space of all continuous functions ''f'' : ''E'' → ''M'': i.e., for every (fixed) ''t'' in ''E'',

			:<math>d(f(s), f(t)) \to 0</math> as <math>\| s - t \| \to 0.</math>

			In almost all applications, one takes ''E'' = [0, ''T''] or [0, +∞) and ''M'' = '''R'''<sup>''n''</sup> for some ''n'' in '''N'''. For brevity, write ''C'' for ''C''([0, ''T'']; '''R'''<sup>''n''</sup>); this is a [[vector space]]. Write ''C''<sub>0</sub> for the [[linear subspace]] consisting only of those functions that take the value zero at the infimum of the set ''E''. Many authors refer to ''C''<sub>0</sub> as "classical Wiener space".

			==Properties of classical Wiener space==
			===Uniform topology===
			The vector space ''C'' can be equipped with the [[uniform norm]]

			:<math>\\| f \\| := \sup_{t \in [0, T]} \| f(t) \|</math>

			turning it into a [[normed vector space]] (in fact a [[Banach space]]). This norm induces a [[metric (mathematics)\|metric]] on ''C'' in the usual way: <math>d (f, g) := \\| f-g \\|</math>. The [[topology]] generated by the [[open set]]s in this metric is the topology of [[uniform convergence]] on [0, ''T''], or the [[uniform topology]].

			Thinking of the domain [0, ''T''] as "time" and the range '''R'''<sup>''n''</sup> as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space a bit" and get the graph of ''f'' to lie on top of the graph of ''g'', while leaving time fixed. Contrast this with the [[Càdlàg#Skorokhod space\|Skorokhod topology]], which allows us to "wiggle" both space and time.

			===Separability and completeness===
			With respect to the uniform metric, ''C'' is both a [[separable space\|separable]] and a [[complete space]]:
			* separability is a consequence of the [[Stone-Weierstrass theorem]];
			* completeness is a consequence of the fact that the uniform limit of a sequence of continuous functions is itself continuous.

			Since it is both separable and complete, ''C'' is a [[Polish space]].

			===Tightness in classical Wiener space===
			Recall that the [[modulus of continuity]] for a function ''f'' : [0, ''T''] → '''R'''<sup>''n''</sup> is defined by

			:<math>\omega_{f} (\delta) := \sup \left\{ \| f(s) - f(t) \| \left\| s, t \in [0, T], \| s - t \| \leq \delta \right. \right\}.</math>

			This definition makes sense even if ''f'' is not continuous, and it can be shown that ''f'' is continuous [[if and only if]] its modulus of continuity tends to zero as δ → 0:

			:<math>f \in C \iff \omega_{f} (\delta) \to 0</math> as δ → 0.

			By an application of the [[Arzelà-Ascoli theorem]], one can show that a sequence <math>(\mu_{n})_{n = 1}^{\infty}</math> of [[probability measure]]s on classical Wiener space ''C'' is [[tightness of measures\|tight]] if and only if both the following conditions are met:

			:<math>\lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n} \{ f \in C \| \| f(0) \| \geq a \} = 0,</math> and
			:<math>\lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n} \{ f \in C \| \omega_{f} (\delta) \geq \varepsilon \} = 0</math> for all ε > 0.

			===Classical Wiener measure===
			There is a "standard" measure on ''C''<sub>0</sub>, known as '''classical Wiener measure''' (or simply '''Wiener measure'''). Wiener measure has (at least) two equivalent characterizations:

			If one defines [[Brownian motion]] to be a [[Markov property\|Markov]] [[stochastic process]] ''B'' : [0, ''T''] × Ω → '''R'''<sup>''n''</sup>, starting at the origin, with [[almost surely]] continuous paths and independent increments

			:<math>B_{t} - B_{s} \sim \mathrm{Normal} \left( 0, \| t - s \| \right),</math>

			then classical Wiener measure γ is the [[law (stochastic processes)\|law]] of the process ''B''.

			Alternatively, one may use the [[abstract Wiener space]] construction, in which classical Wiener measure γ is the [[Radonifying function\|radonification]] of the [[Cylinder set measure#Cylinder set measures on Hilbert spaces\|canonical Gaussian cylinder set measure]] on the Cameron-Martin [[Hilbert space]] corresponding to ''C''<sub>0</sub>.

			Classical Wiener measure is a [[Gaussian measure]]: in particular, it is a [[strictly positive measure\|strictly positive]] probability measure.

			Given classical Wiener measure γ on ''C''<sub>0</sub>, the [[product measure]] γ<sup>''n''</sup> × γ is a probability measure on ''C'', where γ<sup>''n''</sup> denotes the standard [[Gaussian measure]] on '''R'''<sup>''n''</sup>.

			==See also==
			* [[Càdlàg\|Skorokhod space]], a generalization of classical Wiener space, which allows functions to be discontinuous
			* [[Abstract Wiener space]]
			* [[Wiener process]]

			==References==
			* {{cite book \| author=Billingsley, Patrick \| title=Probability and Measure \| publisher=John Wiley & Sons, Inc., New York \| year=1995 \| isbn=0-471-00710-2}}

			[[Category:Stochastic processes]]
			[[Category:Metric geometry]]

en>RjwilmsiBot: Adding Persondata using AWB (7876)

2011-12-12T11:23:31Z

Adding Persondata using AWB (7876)

New page

The author is known by the name of Figures Lint. Playing baseball is the pastime he will by no means quit doing. Bookkeeping is what I do. Years ago he moved to North Dakota and his family enjoys it.<br><br>Here is my site - home std test kit ([https://www.epapyrus.com/xe/Purchase/5258960 hop over to these guys])