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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".
This theorem can be used to disprove the central limit theorem holds for by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for X k {\displaystyle X_{k}} .
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This section illustrates the central limit theorem via an example for which the computation can be done quickly by hand on paper, unlike the more computing-intensive example of the previous section. Sum of all permutations of length 1 selected from the set of integers 1, 2, 3
The central limit theorem can provide more detailed information about the behavior of than the law of large numbers. For example, we can approximately find a tail probability of M N {\displaystyle M_{N}} – the probability that M N {\displaystyle M_{N}} is greater than some value x {\displaystyle x} – for a fixed value of N {\displaystyle N} .
The means and variances of directional quantities are all finite, so that the central limit theorem may be applied to the particular case of directional statistics. [2] This article will deal only with unit vectors in 2-dimensional space (R 2) but the method described can be extended to the general case.
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)
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.