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Difference between revisions of "Cramér theorem"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cramér, "Sur un nouveau théorème-limite de la théorie des probabilités" , ''Act. Sci. et Ind.'' , '''736''' , Hermann (1938) {{MR|}} {{ZBL|64.0529.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) {{MR|0322926}} {{ZBL|0219.60027}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}} </TD></TR></table>
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|valign="top"|{{Ref|C}}|| H. Cramér, "Sur un nouveau théorème-limite de la théorie des probabilités" , ''Act. Sci. et Ind.'' , '''736''' , Hermann (1938) {{MR|}} {{ZBL|64.0529.01}}
 
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|valign="top"|{{Ref|IL}}|| I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) {{MR|0322926}} {{ZBL|0219.60027}}
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|valign="top"|{{Ref|P}}|| V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}}
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) {{MR|0793553}} {{ZBL|0566.60097}} </TD></TR></table>
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|valign="top"|{{Ref|E}}|| R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) {{MR|0793553}} {{ZBL|0566.60097}}
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Revision as of 14:50, 11 May 2012

2020 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
How to Cite This Entry:
Cramér theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cram%C3%A9r_theorem&oldid=26397
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article