Central limit theorem

The central limit theorems are a family of weak convergence results from probability theory. Common to all is the statement that the sum of a large number of independent random variables asymptotically follows a stable distribution. At finite and positive variance of the random variables, the sum is approximately normally distributed, which explains the special position of the normal distribution.

The most important and best-known statement is also referred to simply as The Central Limit Theorem and deals with independent, identically distributed random variables whose expectation and variance are finite. This statement is also known as limit theorem of Lindeberg / Lévy.

There are various generalizations, for which an equal distribution is not a necessary condition. Instead, then another condition is required, which ensures that none of the variables too much influence on the result obtains. Examples are the Lindeberg condition and the Lyapunov condition. Any further generalizations allow even " weak" dependence of the random variables.

The name goes back to G. Polya work via the central limit theorem of probability theory and the problem of moments of 1920.

The Central Limit Theorem of statistics with identical distribution

Let be a sequence of random variables which have the same probability space all have the same distribution and are independent ( iid = independent and identically distributed, Eng. Iid = independent and identically distributed ). Further assume that both the mean value and standard deviation and finally there is.

Now consider the -th partial sum of these random variables. Is the expected value of the variance and is. If one forms from the standardized random variable

Then the Central Limit Theorem states that the distribution function of converges pointwise to the distribution function of the standard normal distribution. Is the distribution function, this means that for each real

In a slightly different notation you get

In which

The average of the first n summands of the random variable is.


  • The proof of the Central Limit Theorem is usually done on the basis of general propositions about the properties of characteristic functions. On this basis, it is sufficient to determine the moments or cumulants of followers and then the coefficients of the Taylor series of the characteristic function. The latter is easily possible ( see Article cumulant, Section Central Limit Theorem ).
  • The Central Limit Theorem but can also elementary, that is, without the deep resources of the characteristic function, are proved. To expected values ​​of the form to be examined which correspond to a hand, in the case of an indicator function of the probability of a completed interval and can be a good approximation in cases of a sufficiently smooth function on the other. This process of an elementary proof comes from Jarl Waldemar Lindeberg.
  • Finite sample sizes leave the question of the convergence quality ascend. Under certain conditions, the set of Berry - Esseen provides an answer: Does the third centered moment and is finite, then the convergence to the normal distribution is uniform and the rate of convergence at least of the order.
  • As for stochastically independent normally distributed random variables, the sum is distributed normally again, the central limit theorem holds for this finite, more precisely, for each after N (0,1) distributed.
  • For stochastically independent Bernoulli - distributed random variables is the sum of a binomial distribution and is obtained as a special case of the central limit theorem the set of Moivre - Laplace.