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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 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
Then according to the central limit theorem, the distribution of Z n approaches the normal N(0, 1 / 3 ) distribution. This convergence is shown in the picture: as n grows larger, the shape of the probability density function gets closer and closer to the Gaussian curve.
The central limit theorem implies that those statistical parameters will have asymptotically normal distributions. The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
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}} .
An important example when the local asymptotic normality holds is in the case of independent and identically distributed sampling from a regular parametric model; this is just the central limit theorem. Barndorff-Nielson & Cox provide a direct definition of asymptotic normality. [2]
The i.i.d. assumption is also used in the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution. [4] The i.i.d. assumption frequently arises in the context of sequences of random variables. Then, "independent and identically ...
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} .