Cramér theorem
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.
References
[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 |
Comments
See also Limit theorems; Probability of large deviations.
References
[E] | R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) MR0793553 Zbl 0566.60097 |
Cramér theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cram%C3%A9r_theorem&oldid=26397