View source for Cramér theorem

{{MSC|60F10}}

[[Category:Limit theorems]] 

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>

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

<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>

<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>

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.

====References====
{|
|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}}
|-
|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}}
|-
|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}}
|}

====Comments====
See also [[Limit theorems|Limit theorems]]; [[Probability of large deviations|Probability of large deviations]].

====References====
{|
|valign="top"|{{Ref|E}}|| R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) {{MR|0793553}} {{ZBL|0566.60097}}
|}