Central limit theorem: the cornerstone of modern statistics

Central limit theorem: the cornerstone of modern statistics

According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ2, distribute normally with mean, µ, and variance, [Formula: see text]. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-par...

Statistics 100 B Instructor : Nicolas Christou The Central Limit Theorem

The central limit theorem states that the sample mean X̄ follows approximately the normal distribution with mean μ and standard deviation σ √ n , where μ and σ are the mean and standard deviation of the population from where the sample was selected. The sample size n has to be large (usually n ≥ 30) if the population from where the sample is taken is nonnormal. If the population follows the norm...

Generalized Central Limit Theorem and Extreme Value Statistics

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

The Martingale Central Limit Theorem

One of the most useful generalizations of the central limit theorem is the martingale central limit theorem of Paul Lévy. Lévy was in part inspired by Lindeberg’s treatment of the central limit theorem for sums of independent – but not necessarily identically distributed – random variables. Lindeberg formulated what, in retrospect, is the right hypothesis, now known as the Lindeberg condition,1...