Central limit theorem from Lindeberg-Feller

The central limit theorem of Lindeberg-Feller , also limit distribution set of Lindeberg-Feller called, is a mathematical theorem of probability theory . It belongs to the central limit value theorems and thus also the limit value theorem of stochastics and is a generalization of the central limit value theorem by Lindeberg-Lévy . This means that under certain conditions the normalized mean values of random variables in distribution converge to the standard normal distribution. Lindeberg-Feller's central limit value theorem weakens these prerequisites by resorting to schemes of random variables in which a certain degree of stochastic dependency between the random variables is allowed. The set is named after Jarl Waldemar Lindeberg and William Feller . Sometimes the sentence is broken down into its sub-statements. One implication is then referred to as Lindeberg's theorem or Lindeberg's Central Limit Distribution Theorem , the other as Feller's theorem .

Framework

In the usual limit theorem, it is always required that the sequence of random variables under consideration is stochastically independent random variables and that the variances are finite. In the case of weaker formulations, it is also required that the random variables are distributed identically. This requirement can, however, be replaced by the Lindeberg condition (for consequences) and the Lyapunov condition (for consequences). ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

Now the question arises whether the requirements for the sentence can be further weakened and a certain degree of dependency is possible. This question can be answered positively. For this purpose, a so-called scheme of random variables is defined . This corresponds to a sequence of small groups of random variables. Each of these small groups of random variables has elements. Formally, schemes of random variables are defined as a sequence of these small groups using double indices. It can now be shown that (under certain further conditions) it is sufficient for convergence to require the independence of the random variables only within the small groups. The relationships of the random variables between different small groups do not matter. ${\displaystyle k_{n}}$

statement

Given a scheme of random variables and be ${\displaystyle (X_{n,l})}$

${\displaystyle S_{n}:=\sum _{l=1}^{k_{n}}X_{n,l}}$.

In addition, it is a standardized , centered and independent scheme . ${\displaystyle (X_{n,l})}$

Then the Lindeberg condition holds (for schemes of random variables) if and only if an asymptotically negligible scheme is and the distribution of in distribution converges to the standard normal distribution, so it holds. ${\displaystyle (X_{n,l})}$ ${\displaystyle S_{n}}$ ${\displaystyle {\mathcal {N}}_{0,1}}$ ${\displaystyle S_{n}{\stackrel {n\to \infty }{\implies }}{\mathcal {N}}_{0,1}}$

comment

Since the Lindeberg condition follows from the Lyapunov condition (for schemes), one can deduce from the Lyapunov condition the convergence in distribution and the asymptotic negligibility. The reverse direction is generally wrong, since the Lyapunov condition does not necessarily follow from the Lindeberg condition.

history

The proof of the theorem followed in two parts. Each part of an implication corresponded to the equivalence formulated above. The conclusion from the Lindeberg condition to the convergence in distribution and the asymptotic negligibility was shown by Jarl Waldemar Lindeberg in 1922. This part is usually more interesting for the applications and partly bears the independent name Lindeberg theorem . The reverse direction ( Feller's Theorem ) was then proven by William Feller in 1935 and 1937.

Web links

• VV Petrov: Lindeberg-Feller theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Lindeberg-Feller Central Limit Theorem . In: MathWorld (English).

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 328.
2. ↑ Kusolitsch: Measure and probability theory. 2014, p. 307.
3. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 204.
4. ↑ Kusolitsch: Measure and probability theory. 2014, p. 311.
5. ↑ Kusolitsch: Measure and probability theory. 2014, p. 312.