Cramér theorem
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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
| [1] | H. Cramér, "Sur un nouveau théorème-limite de la théorie des probabilités" , Act. Sci. et Ind. , 736 , Hermann (1938) MR Template:ZBL |
| [2] | I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian) MR0322926 Template:ZBL |
| [3] | V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Template:ZBL Template:ZBL |
Comments
See also Limit theorems; Probability of large deviations.
References
| [4] | R.S. Ellis, "Entropy, large deviations, and statistical mechanics" , Springer (1985) MR0793553 Template:ZBL |
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