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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".
[4] [5] Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases.
The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used.
The first term on the right-hand-side asymptotically converges to zero, while the second term is qualitatively similar to the summation formula for the central limit theorem in the simpler case of i.i.d. random variables. While the terms in the above expression are not necessarily i.i.d., they are uncorrelated and have zero mean. Indeed:
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
For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the Berry–Esseen theorem. For example, the distributions with finite first, second, and third moment from the exponential family; on the other hand, for some random variables of the heavy tail and fat tail variety, it works very slowly ...
By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal.
The convergence of a random walk toward the Wiener process is controlled by the central limit theorem, and by Donsker's theorem. For a particle in a known fixed position at t = 0, the central limit theorem tells us that after a large number of independent steps in the random walk, the walker's position is distributed according to a normal ...