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