Lindeberg condition

The Lindeberg condition is a term from stochastics . If a sequence of stochastically independent random variables fulfills this condition, the central limit theorem applies to it , even if the random variables are not necessarily distributed identically. In a more general way, the Lindeberg condition can also be formulated for schemes of random variables , in which case a certain degree of dependency between the random variables is permitted. This formulation plays an important role in Lindeberg-Feller's central limit theorem , a generalization of the "ordinary" central limit theorem.

The Lindeberg condition was named after the Finnish mathematician Jarl Waldemar Lindeberg . Another sufficient condition for the central limit theorem is the Lyapunov condition .

Formulation for sequences of random variables

Let be independent, quadratically integrable random variables with for all and be ${\displaystyle X_{1},X_{2},X_{3},\ldots }$ ${\displaystyle {\sigma _{n}}^{2}:={\mbox{Var}}(X_{n})>0}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle s_{n}:={\sqrt {\sum _{k=1}^{n}{\sigma _{k}}^{2}}}\quad ,\quad \mu _{n}:={\mbox{E}}(X_{n})}$.

The Lindeberg condition then applies

${\displaystyle \forall \ \varepsilon >0:\quad \lim _{n\to \infty }\sum _{i=1}^{n}{\mbox{E}}\left({\frac {(X_{i}-\mu _{i})^{2}}{s_{n}^{2}}}\cdot 1_{\left\{|X_{i}-\mu _{i}|>\varepsilon s_{n}\right\}}\right)=\lim _{n\to \infty }{\frac {1}{s_{n}^{2}}}\sum _{i=1}^{n}\int _{\left\{|x-\mu _{i}|>\varepsilon s_{n}\right\}}(x-\mu _{i})^{2}P_{X_{i}}(dx)=0}$ ,

where the indicator function denotes, the sequence satisfies the central limit theorem , i.e. H. the size ${\displaystyle 1_{T}}$ ${\displaystyle (X_{i})_{i}}$

${\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}(X_{i}-\mu _{i})}$

converges in distribution for against a standard normally distributed random variable , i.e. ${\displaystyle n\to \infty }$ ${\displaystyle Z\sim {\mathcal {N}}(0,1)}$

${\displaystyle \forall \ z\in \mathbb {R} :\quad \lim _{n\rightarrow \infty }{\mbox{P}}\left({\frac {1}{s_{n}}}\sum _{i=1}^{n}(X_{i}-\mu _{i})\leq z\right)={\mbox{P}}(Z\leq z)=\Phi (z)}$ ,

where here describes the distribution function of the standard normal distribution . ${\displaystyle \Phi }$

reversal

The reverse of the above is true i. A. not. For this, an additional requirement is necessary for the consequence : ${\displaystyle X_{1},X_{2},\dots }$

Let the independent sequence of square integrable, real random variables with suffice the central limit theorem and further fulfill the Feller-Lévy condition ${\displaystyle (X_{i})_{i}}$ ${\displaystyle \sigma _{i}^{2}>0\ \forall i}$

${\displaystyle \lim _{n\to \infty }\left(\max _{j\in \{1,...,n\}}{\frac {\sigma _{j}}{s_{n}}}\right)=0}$ .

Then the sequence also fulfills the Lindeberg condition . ${\displaystyle (X_{i})_{i}}$

Formulation for schemes of random variables

A centered scheme of random variables is given , in which every random variable can be and are square-integrable ${\displaystyle (X_{n,l}),n\in \mathbb {N} ,l=1,\ldots ,k_{n}}$ ${\displaystyle X_{n,l}}$

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

the sums over the second indices. The scheme now fulfills the Lindeberg condition if it holds for each that ${\displaystyle \varepsilon >0}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{\operatorname {Var} (S_{n})}}\sum _{l=1}^{k_{n}}\operatorname {E} \left(X_{n,l}^{2}\chi _{\{X_{n,l}^{2}>\varepsilon ^{2}\operatorname {Var} (S_{n})\}}\right)=0}$

is.

literature

• Heinz Bauer : Probability Theory . De Gruyter, Berlin / New York 2002, ISBN 3110172364 , p. 239.
• JW Lindeberg: A new derivation of the exponential law in probability theory . In: Mathematische Zeitschrift , Volume 15, 1922, pp. 211–225.
• 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 .

Web link

• Eric W. Weisstein: Lindeberg Condition. on MathWorld.

Individual evidence

1. ↑ Eric W. Weisstein : Feller-Lévy Condition . In: MathWorld (English).