Cramér's decomposition theorem

In mathematical statistics, Cramér's theorem (or Cramér’s decomposition theorem) is one of several theorems of Harald Cramér, a Swedish statistician and probabilist.

Normal random variables

Cramér's theorem is the result that if X and Y are independent real-valued random variables whose sum X + Y is a normal random variable, then both X and Y must be normal as well. By 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 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 independent, identically distributed random variables, say X₁, X₂, X₃, …. It is well known, by the law of large numbers, that in this case the sequence

\left({\frac {\sum _{k=1}^{n}X_{k}}{n}}\right)_{n\in \mathbb {N} }

converges in probability to the mean of the probability distribution of X_k. Cramér's theorem in this sense states that the probabilities of "large deviations" away from the mean in this sequence decay exponentially with the rate given by the Cramér function, which is the Legendre transform of the cumulant-generating function of X_k.

Slutsky's theorem

Slutsky’s theorem is also attributed to Harald Cramér.^[1] This theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.

Notes

^ Slutsky's theorem is also called Cramér’s theorem according to Remark 11.1 (page 249) of Allan Gut. A Graduate Course in Probability. Springer Verlag. 2005.

References

Cramér, Harald (1936). "Über eine Eigenschaft der normalen Verteilungsfunktion". Mathematische Zeitschrift (in German). 41 (1): 405–414. doi:10.1007/BF01180430. MR 1545629.
Cramér, Harald (1938). "Sur un nouveau théorème-limite de la théorie des probabilités". Actualités Scientifiques et Industrielles (in French). 736: 5–23.
Fan, X.; Grama, I.; Liu, Q. (2013). "Cramér large deviation expansions for martingales under Bernstein's condition". Stochastic Process. Appl. 123: 3919–3942.
Lukacs, Eugen: Characteristic functions. Griffin, London 1960 (2. Edition 1970), ISBN 0-85264-170-2.

External links

"Cramér theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Cramér's theorem". MathWorld.

[1] Slutsky's theorem is also called Cramér’s theorem according to Remark 11.1 (page 249) of Allan Gut. A Graduate Course in Probability. Springer Verlag. 2005.

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