Cramér theorem

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2010 Mathematics Subject Classification: Primary: 60F10 [MSN][ZBL]

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.


[C] H. Cramér, "Sur un nouveau théorème-limite de la théorie des probabilités" , Act. Sci. et Ind. , 736 , Hermann (1938) Zbl 64.0529.01
[IL] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) MR0322926 Zbl 0219.60027
[P] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042


See also Limit theorems; Probability of large deviations.


[E] R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) MR0793553 Zbl 0566.60097
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