Lindeberg's condition

The Lindeberg condition is a term from the stochastic. Meets a sequence of stochastically independent random variables this condition shall apply the Central Limit Theorem for them, even if the random variables are not necessarily identically distributed as.

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

Formulation

Be independent, square-integrable random variables with for all and be

Then applies the Lindeberg condition

The result so satisfies the central limit theorem, ie the size

Converges in distribution for against a standard normal distributed random variable, ie

Wherein here describes the distribution function of the standard normal distribution.

Reversal

The converse of the above facts shall I.A. do not. For this purpose, an additional requirement for the sequence is required:

The independent sequence of square-integrable, real random variables with sufficient the central limit theorem and fulfill further the Feller condition

Then fulfills the follow the Lindeberg condition.