Cramér theorem

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An integral limit theorem for the probability of large deviations of sums of independent random variables. Let be a sequence of independent random variables with the same non-degenerate distribution function , such that and such that the generating function of the moments is finite in some interval (this last condition is known as the Cramér condition). Let

If , as , then

Here is the normal distribution function and is the so-called Cramér series, the coefficients of which depend only on the moments of the random variable ; this series is convergent for all sufficiently small . Actually, the original result, obtained by H. Cramér in 1938, was somewhat weaker than that just described.


[1] H. Cramér, "Sur un nouveau théorème-limite de la théorie des probabilités" , Act. Sci. et Ind. , 736 , Hermann (1938)
[2] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)
[3] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)


See also Limit theorems; Probability of large deviations.


[a1] R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985)
How to Cite This Entry:
Cramér theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article