Set of berry lakes

The Berry-Esseen theorem is a proposition from the theory of probability , the statements about the quality of convergence in the central limit theorem is true. It gives both the speed of convergence and a numerical estimate for the approximation to the normal distribution. The theorem was independently proven by mathematicians Andrew C. Berry (1941) and Carl-Gustav Esseen (1942, published 1944).

Set of berry lakes

Let it be a sequence of independent and identically distributed random variables on a probability space , for which the expectation values and the variances exist and are finite. Then the distribution functions converge according to the central limit theorem ${\displaystyle (X_{n})}$ ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle \mu =\mathrm {E} (X_{n})}$ ${\displaystyle \sigma ^{2}=\mathrm {E} ((X_{n}-\mu )^{2})}$

${\displaystyle F_{n}(x)=P\left({\frac {X_{1}+\ldots +X_{n}-n\cdot \mu }{\sigma \cdot {\sqrt {n}}}}\leq x\right)}$ For ${\displaystyle x\in \mathbb {R} }$

the standardized sums against the normal distribution . ${\displaystyle \Phi _{0,1}}$

If the third absolute moment of the random variable exists, then the estimate holds for a general constant that is independent of the distribution of the random variable ${\displaystyle \rho =\mathrm {E} (|X_{n}-\mu |^{3})}$ ${\displaystyle X_{n}}$ ${\displaystyle C}$ ${\displaystyle X_{n}}$

${\displaystyle \left|F_{n}(x)-\Phi _{0,1}(x)\right|\leq {\frac {C\cdot \rho }{\sigma ^{3}{\sqrt {n}}}}}$for everyone . ${\displaystyle x\in \mathbb {R} }$

In the event that they are distributed independently but not identically, with , and , and the estimate ${\displaystyle X_{1},X_{2},\ldots }$ ${\displaystyle \mu _{i}=\mathrm {E} (X_{i})}$ ${\displaystyle \sigma _{i}^{2}=\mathrm {E} ((X_{i}-\mu _{i})^{2})<\infty }$ ${\displaystyle \rho _{i}=\mathrm {E} (|X_{i}-\mu _{i}|^{3})<\infty }$ ${\displaystyle r_{n}=\rho _{1}+\ldots +\rho _{n}}$ ${\displaystyle s_{n}^{2}=\sigma _{1}^{2}+\ldots +\sigma _{n}^{2}}$

${\displaystyle \left|F_{n}(x)-\Phi _{0,1}(x)\right|\leq {\frac {C_{x}\cdot r_{n}}{s_{n}^{3}}}}$for everyone . ${\displaystyle x\in \mathbb {R} }$

Remarks

• For the validity of the Berry-Esseen theorem, in addition to the requirements for the central limit value theorem (existence of expected value and variance), the existence of the third moment is also required. Therefore, the theorem does not provide a statement about the quality of the convergence to the normal distribution for all cases in which the Central Limit Theorem applies.
• The Berry-Esseen theorem gives as a qualitative statement the speed of convergence in the central limit theorem with the order of magnitude . Without further conditions to the distribution of the random variable , this is the best possible order of magnitude as the special case of the Bernoulli distribution with shows. ${\displaystyle 1/{\sqrt {n}}}$ ${\displaystyle X_{n}}$ ${\displaystyle \mathrm {P} \{X_{n}=0\}=\mathrm {P} \{X_{n}=1\}=1/2}$
• The theorem provides a quantitative estimate of the approximation to the normal distribution. The constant is a "universal constant" that does not depend on the properties of the random variable . ${\displaystyle C}$ ${\displaystyle X_{n}}$

The Berry-Esseen constant

The constant that is important for the quantitative estimation of the convergence is referred to in the literature as the Berry-Esseen constant (Berry-Esseen bound). ${\displaystyle C}$

In the original work by Carl-Gustav Esseen, C is given as 7.59. Since then it has been continuously improved. In 1985 Shiganov gave the value C = 0.7655. The best value known to date (as of 2012) is C = 0.4748. On the other hand, it follows from the special case of the Bernoulli distribution mentioned above that must be greater than . Esseen himself proved that is bigger than . ${\displaystyle C}$ ${\displaystyle 1/{\sqrt {2\pi }}\approx 0{,}3989}$ ${\displaystyle C}$ ${\displaystyle {\frac {{\sqrt {10}}+3}{6{\sqrt {2\pi }}}}\approx 0{,}4097}$

In the case of not identical distribution, the following applies . ${\displaystyle C_{x}\leq 0{,}5591}$

literature

• Andrew C. Berry: The accuracy of the Gaussian approximation to the sum of independent variables. In: Transaction of the American Mathematical Society. 49, 1941, pp. 122-136.
• Carl-Gustav Esseen: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Dissertation. In: Acta mathematica. 77, 1944.
• William Feller : An Introduction to Probability Theory and Its Applications. Volume II. John Wiley & Sons, New York 1972, ISBN 0-471-25709-5 .

Individual evidence

1. ↑ ^{a b} I. S. Tyurin: A Refinement of the Remainder in the Lyapunov theorem. In: Theory of Probability and Its Applications. 56, 4, 2012, pp. 693-696 ( doi : 10.1137 / S0040585X9798572X ).
2. ↑ IS Shiganov: Refinement of the upper bound of the constant in the central limit theorem. In: Journal of Soviet Mathematics. 1986, pp. 2545-2550 ( doi : 10.1007 / BF01121471 ).
3. ↑ Irina Shevtsova: On the absolute constants in the Berry – Esseen type inequalities for identically distributed summands. ( online , PDF, 141 kB).
4. ^ Carl-Gustav Esseen: A moment inequality with an application to the central limit theorem. In: Skandinavisk Aktuarietidskrift. 39, 1956, pp. 160-170.
   See also: Berry-Esseen Constant. In: Steven R. Finch: Mathematical Constants. Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , p. 264 ( limited preview in Google book search).