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{{MSC|60F10}}

[[Category:Limit theorems]] 

An integral limit theorem for the probability of large deviations of sums of independent random variables. Let  $  X _ {1} , X _ {2} \dots $
be a sequence of independent random variables with the same non-degenerate distribution function  $  F $, 
such that  $  {\mathsf E} X _ {1} = 0 $
and such that the generating function  $  {\mathsf E} e ^ {tX _ {1} } $
of the moments is finite in some interval  $  | t | < H $(
this last condition is known as the Cramér condition). Let

$$ 
{\mathsf E} X _ {1}  ^ {2}  =  \sigma  ^ {2} ,\ \ 
F _ {n} ( x)  =  {\mathsf P} \left (

\frac{1}{\sigma n  ^ {1/2} }

\sum _ {j = 1 } ^ { n } 
X _ {j} < x \right ) .
$$

If  $  x > 1 $, 
$  x = o ( \sqrt n ) $
as  $  n \rightarrow \infty $, 
then

$$ 

\frac{1 - F _ {n} ( x) }{1 - \Phi ( x) }
  = \ 
 \mathop{\rm exp} \left \{

\frac{x  ^ {3} }{\sqrt n }

\lambda \left ( 
\frac{x}{\sqrt n }
 \right )  \right \}
\left [ 1 + O \left ( 
\frac{x}{\sqrt n }
 \right )  \right ] ,
$$

$$ 

\frac{F _ {n} (- x) }{\Phi (- x) }
  =   \mathop{\rm exp} \left \{ -

\frac{x  ^ {3} }{\sqrt n }
 \lambda \left ( - {
\frac{x}{\sqrt n }
 } \right ) \ 
\right \} \left [ 1 + O \left ( 
\frac{x}{\sqrt n }
 \right )  \right ] .
$$

Here  $  \Phi ( x) $
is the normal  $  ( 0, 1) $
distribution function and  $  \lambda ( t) = \sum _ {k = 0 }   ^  \infty  c _ {k} t  ^ {k} $
is the so-called Cramér series, the coefficients of which depend only on the moments of the random variable  $  X _ {1} $; 
this series is convergent for all sufficiently small  $  t $. 
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}}
|}