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An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables.
Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric.It was introduced by Charles Stein, who first published it in 1972, [1] to obtain a bound between the distribution of a sum of -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform ...
Animated examples of the CLT; General Dynamic SOCR CLT Activity; Interactive Simulation of the Central Limit Theorem for Windows; The SOCR CLT activity provides hands-on demonstration of the theory and applications of this limit theorem. A music video demonstrating the central limit theorem with a Galton board by Carl McTague
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The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
Cayley's theorem (group theory) Central limit theorem (probability) Cesàro's theorem (real analysis) Ceva's theorem ; Chasles' theorem, Chasles' theorem ; Chasles' theorem (algebraic geometry) Chebotarev's density theorem (number theory) Chen's theorem (number theory) Cheng's eigenvalue comparison theorem (Riemannian geometry)
This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate X —for which the mean and variance of ln( X ) are specified.