limit theorem for convex bodies". Another example: where and . If then factorizes into which means are independent. In general, however, they are dependent. The condition ensures that are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent. By the way, pairwise independence cannot replace independence in the classical central limit theorem. Here is a Berry–Esseen type result. The distribution of need not be approximately normal (in fact, it can be uniform). However, the distribution of is close to (in the total variation distance) for most vectors according to the uniform distribution on the sphere . Lacunary trigonometric series Gaussian polytopes The same also holds in all dimensions greater than 2. The polytope is called a Gaussian random polytope. A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions. Linear functions of orthogonal matrices A linear function of a matrix is a linear combination of its elements (with given coefficients), where is the matrix of the coefficients; see Trace (linear algebra)#Inner product. A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group ; see Rotation matrix#Uniform random rotation matrices. Subsequences Random walk on a crystal lattice The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures. Applications and examples A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments. Regression Regression analysis and in particular ordinary least squares specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive error term. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if