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 .
The reverse of the above is true i. A. not. For this, an additional requirement is necessary for the consequence :
Let the independent sequence of square integrable, real random variables with suffice the central limit theorem and further fulfill the Feller-Lévy condition
.
Then the sequence also fulfills the Lindeberg condition .
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
the sums over the second indices. The scheme now fulfills the Lindeberg condition if it holds for each that
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 .