Quantum spin Hall effect - Revision history

76.24.26.202 at 01:02, 9 October 2014

2014-10-09T01:02:10Z

← Older revision		Revision as of 03:02, 9 October 2014
Line 1:		Line 1:
	~~In [[mathematical statistics]], '''Cramér's theorem''' (or '''Cramér’s decomposition theorem''')~~ is ~~one of several theorems of [[Harald Cramér]], a [[Sweden\|Swedish]] [[statistician]]~~ and ~~[[probabilist]].~~		Ed is what people contact me and my wife doesn't like it at all. To perform lacross is something he would by no means give up. I am presently a travel agent. Alaska is where I've usually been residing.<br><br>Check out my web page ... [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review real psychic]

	~~== Normal random variables ==~~
	~~Cramér~~'s theorem is the result that if ''X'' and ''Y'' are [[independence (probability theory)\|independent]] [[real line\|real-valued]] [[random variable]]s whose sum ''X'' + ''Y'' is a [[normal distribution\|normal random variable]], then both ''X'' and ''Y'' must be normal as well. By [[mathematical induction\|induction]], if any finite sum of independent real-valued random variables is normal, then the summands must all ~~be normal~~.

	~~Thus, while the normal distribution~~ is ~~[[Infinite divisibility (probability)\|infinitely divisible]], it can ''only'' be decomposed into normal distributions (if the summands are independent)~~.

	Contrast with the [[central limit theorem]], which states that the average of independent identically distributed random variables with finite mean and variance is ''asymptotically'' normal. Cramér's theorem shows that a ~~finite average is not normal, unless the original variables were normal.~~

	~~== Large deviations ==~~
	'''Cramér's theorem''' may also refer to another result of the same mathematician concerning the partial sums of a sequence of [[iid\|independent, identically distributed]] random variables, say ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, …. It is ~~well known, by the [[law of large numbers]], that in this case the sequence~~

	~~:<math>\left(\frac{\sum_{k=1}^n X_k}{n}\right)_{n\in \mathbb N}</math>~~

	~~converges in probability to the [[mean]] of the probability distribution of ''X<sub>k</sub>'~~'. Cramér's theorem in this sense states that the probabilities of "[[Large deviations theory\|large deviations]]" away from the mean in this sequence [[exponential decay\|decay exponentially]] with the '''[[Rate function\|rate]]''' given by the ''Cramér function'', which is the [[Legendre transformation\|Legendre transform]] of the [[cumulant]]-generating function of ''X<~~sub~~>~~k</sub>''.~~

	~~==Slutsky's theorem==~~
	~~[[Slutsky’s theorem]] is also attributed to [[Harald Cramér]].~~<~~ref~~>~~[[Slutsky's theorem]] is also called [[Harald Cramér\|Cramér]]’s theorem according to Remark 11.1 (~~page ~~249) of Allan Gut. ''A Graduate Course in Probability~~.~~'' Springer Verlag~~. ~~2005~~.</~~ref> This theorem extends some properties of algebraic operations on [[Limit of a sequence\|convergent sequences]] of [[real number]]s to sequences of [[random variable]]s.~~

	~~== See also ==~~
	* [[Asymptotic equipartition property]]
	* [[Cochran's theorem]], on decomposing sum of squares of normal distributions
	* [[Indecomposable distribution]], on decomposability
	* [[Raikov's theorem]], on the decomposition of Poisson distributions
	* [[Infinite divisibility (probability)]]

	~~==References==~~
	~~<references~~/>

	* {{cite journal
	~~\| last = Cramér~~
	~~\| first = Harald~~
	~~\| authorlink=Harald Cramér~~
	~~\| title = Über eine Eigenschaft der normalen Verteilungsfunktion~~
	~~\| journal = Mathematische Zeitschrift~~
	~~\| volume = 41~~
	~~\| year = 1936~~
	~~\| issue = 1~~
	~~\| pages = 405–414~~
	~~\| language = German~~
	~~\| doi = 10~~.~~1007~~/~~BF01180430~~
	~~\| mr = 1545629~~
	}}
	* {{cite journal
	~~\| last = Cramér~~
	~~\| first = Harald~~
	~~\| authorlink=Harald Cramér~~
	~~\| title = Sur un nouveau théorème-limite de la théorie des probabilités~~
	~~\| journal = Actualités Scientifiques et Industrielles~~
	~~\| volume = 736~~
	~~\| year = 1938~~
	~~\| pages = 5–23~~
	~~\| language = French~~
	}}
	* {{cite journal
	~~\| last1 = Fan~~
	~~\| first1 = X~~.
	~~\| last2~~ = ~~Grama~~
	~~\| first2~~ = I.
	~~\| last3 = Liu~~
	~~\| first3 = Q.~~
	~~\| title = Cramér large deviation expansions for martingales under Bernstein's condition~~
	~~\| journal = Stochastic Process. Appl.~~
	~~\| volume = 123~~
	~~\| year = 2013~~
	~~\| pages = 3919–3942~~
	}}

	*Lukacs, Eugen: ''Characteristic functions''. Griffin, London 1960 (2. Edition 1970), ISBN 0-85264-170-2.

	~~==External links==~~
	* {{MathWorld\|urlname=CramersTheorem\|title=Cramér's theorem}}

	~~{{DEFAULTSORT:Cramers theorem}}~~
	~~[[Category:Probability theorems]]~~
	~~[[Category:Statistical theorems]]~~
	~~[[Category:Characterization of probability distributions]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7269101

2013-03-17T16:01:47Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7269101

← Older revision		Revision as of 18:01, 17 March 2013
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	~~The individual who wrote~~ the ~~article~~ is ~~known~~ as ~~Jayson Hirano and he completely digs that title~~. ~~Mississippi~~ is ~~where his house~~ is. ~~To climb~~ is ~~some thing she would by no means give up~~. ~~Credit authorising~~ is ~~how she makes~~ a ~~living.~~<br><br>~~My web blog; spirit messages -~~ [~~http~~://~~formalarmour~~.~~com~~/~~index~~.~~php?do~~=/~~profile-26947~~/~~info~~/ ~~her response~~] -		In [[mathematical statistics]], '''Cramér's theorem''' (or '''Cramér’s decomposition theorem''') is one of several theorems of [[Harald Cramér]], a [[Sweden\|Swedish]] [[statistician]] and [[probabilist]].

			== Normal random variables ==
			Cramér's theorem is the result that if ''X'' and ''Y'' are [[independence (probability theory)\|independent]] [[real line\|real-valued]] [[random variable]]s whose sum ''X'' + ''Y'' is a [[normal distribution\|normal random variable]], then both ''X'' and ''Y'' must be normal as well. By [[mathematical induction\|induction]], if any finite sum of independent real-valued random variables is normal, then the summands must all be normal.

			Thus, while the normal distribution is [[Infinite divisibility (probability)\|infinitely divisible]], it can ''only'' be decomposed into normal distributions (if the summands are independent).

			Contrast with the [[central limit theorem]], which states that the average of independent identically distributed random variables with finite mean and variance is ''asymptotically'' normal. Cramér's theorem shows that a finite average is not normal, unless the original variables were normal.

			== Large deviations ==
			'''Cramér's theorem''' may also refer to another result of the same mathematician concerning the partial sums of a sequence of [[iid\|independent, identically distributed]] random variables, say ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, …. It is well known, by the [[law of large numbers]], that in this case the sequence

			:<math>\left(\frac{\sum_{k=1}^n X_k}{n}\right)_{n\in \mathbb N}</math>

			converges in probability to the [[mean]] of the probability distribution of ''X<sub>k</sub>''. Cramér's theorem in this sense states that the probabilities of "[[Large deviations theory\|large deviations]]" away from the mean in this sequence [[exponential decay\|decay exponentially]] with the '''[[Rate function\|rate]]''' given by the ''Cramér function'', which is the [[Legendre transformation\|Legendre transform]] of the [[cumulant]]-generating function of ''X<sub>k</sub>''.

			==Slutsky's theorem==
			[[Slutsky’s theorem]] is also attributed to [[Harald Cramér]].<ref>[[Slutsky's theorem]] is also called [[Harald Cramér\|Cramér]]’s theorem according to Remark 11.1 (page 249) of Allan Gut. ''A Graduate Course in Probability.'' Springer Verlag. 2005.</ref> This theorem extends some properties of algebraic operations on [[Limit of a sequence\|convergent sequences]] of [[real number]]s to sequences of [[random variable]]s.

			== See also ==
			* [[Asymptotic equipartition property]]
			* [[Cochran's theorem]], on decomposing sum of squares of normal distributions
			* [[Indecomposable distribution]], on decomposability
			* [[Raikov's theorem]], on the decomposition of Poisson distributions
			* [[Infinite divisibility (probability)]]

			==References==
			<references/>

			* {{cite journal
			\| last = Cramér
			\| first = Harald
			\| authorlink=Harald Cramér
			\| title = Über eine Eigenschaft der normalen Verteilungsfunktion
			\| journal = Mathematische Zeitschrift
			\| volume = 41
			\| year = 1936
			\| issue = 1
			\| pages = 405–414
			\| language = German
			\| doi = 10.1007/BF01180430
			\| mr = 1545629
			}}
			* {{cite journal
			\| last = Cramér
			\| first = Harald
			\| authorlink=Harald Cramér
			\| title = Sur un nouveau théorème-limite de la théorie des probabilités
			\| journal = Actualités Scientifiques et Industrielles
			\| volume = 736
			\| year = 1938
			\| pages = 5–23
			\| language = French
			}}
			* {{cite journal
			\| last1 = Fan
			\| first1 = X.
			\| last2 = Grama
			\| first2 = I.
			\| last3 = Liu
			\| first3 = Q.
			\| title = Cramér large deviation expansions for martingales under Bernstein's condition
			\| journal = Stochastic Process. Appl.
			\| volume = 123
			\| year = 2013
			\| pages = 3919–3942
			}}

			*Lukacs, Eugen: ''Characteristic functions''. Griffin, London 1960 (2. Edition 1970), ISBN 0-85264-170-2.

			==External links==
			* {{MathWorld\|urlname=CramersTheorem\|title=Cramér's theorem}}

			{{DEFAULTSORT:Cramers theorem}}
			[[Category:Probability theorems]]
			[[Category:Statistical theorems]]
			[[Category:Characterization of probability distributions]]

166.111.177.133 at 23:45, 14 August 2012

2012-08-14T23:45:40Z

New page

The individual who wrote the article is known as Jayson Hirano and he completely digs that title. Mississippi is where his house is. To climb is some thing she would by no means give up. Credit authorising is how she makes a living.<br><br>My web blog; spirit messages - [http://formalarmour.com/index.php?do=/profile-26947/info/ her response] -

← Older revision		Revision as of 03:02, 9 October 2014
Line 1:		Line 1:
	~~In [[mathematical statistics]], '''Cramér's theorem''' (or '''Cramér’s decomposition theorem''')~~ is ~~one of several theorems of [[Harald Cramér]], a [[Sweden\|Swedish]] [[statistician]]~~ and ~~[[probabilist]].~~		Ed is what people contact me and my wife doesn't like it at all. To perform lacross is something he would by no means give up. I am presently a travel agent. Alaska is where I've usually been residing.<br><br>Check out my web page ... [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review real psychic]

	~~== Normal random variables ==~~
	~~Cramér~~'s theorem is the result that if ''X'' and ''Y'' are [[independence (probability theory)\|independent]] [[real line\|real-valued]] [[random variable]]s whose sum ''X'' + ''Y'' is a [[normal distribution\|normal random variable]], then both ''X'' and ''Y'' must be normal as well. By [[mathematical induction\|induction]], if any finite sum of independent real-valued random variables is normal, then the summands must all ~~be normal~~.

	~~Thus, while the normal distribution~~ is ~~[[Infinite divisibility (probability)\|infinitely divisible]], it can ''only'' be decomposed into normal distributions (if the summands are independent)~~.

	Contrast with the [[central limit theorem]], which states that the average of independent identically distributed random variables with finite mean and variance is ''asymptotically'' normal. Cramér's theorem shows that a ~~finite average is not normal, unless the original variables were normal.~~

	~~== Large deviations ==~~
	'''Cramér's theorem''' may also refer to another result of the same mathematician concerning the partial sums of a sequence of [[iid\|independent, identically distributed]] random variables, say ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, …. It is ~~well known, by the [[law of large numbers]], that in this case the sequence~~

	~~:<math>\left(\frac{\sum_{k=1}^n X_k}{n}\right)_{n\in \mathbb N}</math>~~

	~~converges in probability to the [[mean]] of the probability distribution of ''X<sub>k</sub>'~~'. Cramér's theorem in this sense states that the probabilities of "[[Large deviations theory\|large deviations]]" away from the mean in this sequence [[exponential decay\|decay exponentially]] with the '''[[Rate function\|rate]]''' given by the ''Cramér function'', which is the [[Legendre transformation\|Legendre transform]] of the [[cumulant]]-generating function of ''X<~~sub~~>~~k</sub>''.~~

	~~==Slutsky's theorem==~~
	~~[[Slutsky’s theorem]] is also attributed to [[Harald Cramér]].~~<~~ref~~>~~[[Slutsky's theorem]] is also called [[Harald Cramér\|Cramér]]’s theorem according to Remark 11.1 (~~page ~~249) of Allan Gut. ''A Graduate Course in Probability~~.~~'' Springer Verlag~~. ~~2005~~.</~~ref> This theorem extends some properties of algebraic operations on [[Limit of a sequence\|convergent sequences]] of [[real number]]s to sequences of [[random variable]]s.~~

	~~== See also ==~~
	* [[Asymptotic equipartition property]]
	* [[Cochran's theorem]], on decomposing sum of squares of normal distributions
	* [[Indecomposable distribution]], on decomposability
	* [[Raikov's theorem]], on the decomposition of Poisson distributions
	* [[Infinite divisibility (probability)]]

	~~==References==~~
	~~<references~~/>

	* {{cite journal
	~~\| last = Cramér~~
	~~\| first = Harald~~
	~~\| authorlink=Harald Cramér~~
	~~\| title = Über eine Eigenschaft der normalen Verteilungsfunktion~~
	~~\| journal = Mathematische Zeitschrift~~
	~~\| volume = 41~~
	~~\| year = 1936~~
	~~\| issue = 1~~
	~~\| pages = 405–414~~
	~~\| language = German~~
	~~\| doi = 10~~.~~1007~~/~~BF01180430~~
	~~\| mr = 1545629~~
	}}
	* {{cite journal
	~~\| last = Cramér~~
	~~\| first = Harald~~
	~~\| authorlink=Harald Cramér~~
	~~\| title = Sur un nouveau théorème-limite de la théorie des probabilités~~
	~~\| journal = Actualités Scientifiques et Industrielles~~
	~~\| volume = 736~~
	~~\| year = 1938~~
	~~\| pages = 5–23~~
	~~\| language = French~~
	}}
	* {{cite journal
	~~\| last1 = Fan~~
	~~\| first1 = X~~.
	~~\| last2~~ = ~~Grama~~
	~~\| first2~~ = I.
	~~\| last3 = Liu~~
	~~\| first3 = Q.~~
	~~\| title = Cramér large deviation expansions for martingales under Bernstein's condition~~
	~~\| journal = Stochastic Process. Appl.~~
	~~\| volume = 123~~
	~~\| year = 2013~~
	~~\| pages = 3919–3942~~
	}}

	*Lukacs, Eugen: ''Characteristic functions''. Griffin, London 1960 (2. Edition 1970), ISBN 0-85264-170-2.

	~~==External links==~~
	* {{MathWorld\|urlname=CramersTheorem\|title=Cramér's theorem}}

	~~{{DEFAULTSORT:Cramers theorem}}~~
	~~[[Category:Probability theorems]]~~
	~~[[Category:Statistical theorems]]~~
	~~[[Category:Characterization of probability distributions]~~]