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| 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]].
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| == Normal random variables ==
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| 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.
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| Thus, while the normal distribution is [[Infinite divisibility (probability)|infinitely divisible]], it can ''only'' be decomposed into normal distributions (if the summands are independent).
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| 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.
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| == Large deviations ==
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| '''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
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| :<math>\left(\frac{\sum_{k=1}^n X_k}{n}\right)_{n\in \mathbb N}</math>
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| 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>''.
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| ==Slutsky's theorem==
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| [[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.
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| == See also ==
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| * [[Asymptotic equipartition property]]
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| * [[Cochran's theorem]], on decomposing sum of squares of normal distributions
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| * [[Indecomposable distribution]], on decomposability
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| * [[Raikov's theorem]], on the decomposition of Poisson distributions
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| * [[Infinite divisibility (probability)]]
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| ==References==
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| <references/>
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| * {{cite journal
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| | last = Cramér
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| | first = Harald
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| | authorlink=Harald Cramér
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| | title = Über eine Eigenschaft der normalen Verteilungsfunktion
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| | journal = Mathematische Zeitschrift
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| | volume = 41
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| | year = 1936
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| | issue = 1
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| | pages = 405–414
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| | language = German
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| | doi = 10.1007/BF01180430
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| | mr = 1545629
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| }}
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| * {{cite journal
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| | last = Cramér
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| | first = Harald
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| | authorlink=Harald Cramér
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| | title = Sur un nouveau théorème-limite de la théorie des probabilités
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| | journal = Actualités Scientifiques et Industrielles
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| | volume = 736
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| | year = 1938
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| | pages = 5–23
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| | language = French
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| }}
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| * {{cite journal
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| | last1 = Fan
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| | first1 = X.
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| | last2 = Grama
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| | first2 = I.
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| | last3 = Liu
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| | first3 = Q.
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| | title = Cramér large deviation expansions for martingales under Bernstein's condition
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| | journal = Stochastic Process. Appl.
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| | volume = 123
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| | year = 2013
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| | pages = 3919–3942
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| }}
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| *Lukacs, Eugen: ''Characteristic functions''. Griffin, London 1960 (2. Edition 1970), ISBN 0-85264-170-2.
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| ==External links==
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| * {{MathWorld|urlname=CramersTheorem|title=Cramér's theorem}}
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| {{DEFAULTSORT:Cramers theorem}}
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| [[Category:Probability theorems]]
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| [[Category:Statistical theorems]]
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| [[Category:Characterization of probability distributions]]
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