of the difference between the sample average and its limit when multiplied by the factor approximates the normal distribution with mean 0 and variance For large enough , the distribution of gets arbitrarily close to the normal distribution with mean and variance The usefulness of the theorem is that the distribution of approaches normality regardless of the shape of the distribution of the individual Formally, the theorem can be stated as follows: In the case convergence in distribution means that the cumulative distribution functions of converge pointwise to the cdf of the distribution: for every real where is the standard normal cdf evaluated The convergence is uniform in in the sense that where denotes the least upper bound (or supremum) of the set. Lyapunov CLT The theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the central limit theorem the random variables have to be independent, but not necessarily identically distributed. The theorem also requires that random variables have moments of some order and that the rate of growth of these moments is limited by the Lyapunov condition given below. In practice it is usually easiest to check Lyapunov's condition for If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold. Lindeberg CLT In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from Lindeberg in 1920). Suppose that for every where is the indicator function. Then the distribution of the standardized sums converges towards the standard normal distribution Multidimensional CLT Proofs that use characteristic functions can be extended to cases where each individual is a random vector in with mean vector and covariance matrix (among the components of the vector), and these random vectors are independent and identically distributed. Summation of these vectors is being done component-wise. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution. Let be the -vector. The bold in means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be and the average is and therefore The multivariate central limit theorem states that where the covariance matrix is equal to The rate of convergence is given by the following Berry–Esseen type result: It is unknown whether the factor is necessary. Generalized