Cramér theorem

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 $ 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

[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