Difference between revisions of "Cramér theorem"
From Encyclopedia of Mathematics
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| + | An integral limit theorem for the probability of large deviations of sums of independent random variables. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270001.png" /> be a sequence of independent random variables with the same non-degenerate distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270002.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270003.png" /> and such that the generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270004.png" /> of the moments is finite in some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270005.png" /> (this last condition is known as the Cramér condition). Let | ||
| + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270006.png" /></td> </tr></table> | ||
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| + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270008.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c0270009.png" />, then | ||
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| + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c02700010.png" /></td> </tr></table> | ||
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| + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c02700011.png" /></td> </tr></table> | ||
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| + | Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c02700012.png" /> is the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c02700013.png" /> distribution function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c02700014.png" /> is the so-called Cramér series, the coefficients of which depend only on the moments of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c02700015.png" />; this series is convergent for all sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027000/c02700016.png" />. Actually, the original result, obtained by H. Cramér in 1938, was somewhat weaker than that just described. | ||
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| + | ====References==== | ||
| + | <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)</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)</TD></TR></table> | ||
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| + | ====Comments==== | ||
| + | See also [[Limit theorems|Limit theorems]]; [[Probability of large deviations|Probability of large deviations]]. | ||
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| + | ====References==== | ||
| + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985)</TD></TR></table> | ||
Revision as of 19:21, 7 February 2012
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
| [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) |
Comments
See also Limit theorems; Probability of large deviations.
References
| [a1] | R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) |
How to Cite This Entry:
Cramér theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cram%C3%A9r_theorem&oldid=20876
Cramér theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cram%C3%A9r_theorem&oldid=20876
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article