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A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of Alan Turing's 1934 Fellowship Dissertation for King's College at the University of Cambridge. Only after submitting the work did Turing learn it had already been proved.
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.
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}} .
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
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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.
On the other hand, the central limit theorem states that the sums S n scaled by the factor n −1/2 converge in distribution to a standard normal distribution. By Kolmogorov's zero–one law , for any fixed M , the probability that the event lim sup n S n n ≥ M {\displaystyle \limsup _{n}{\frac {S_{n}}{\sqrt {n}}}\geq M} occurs is 0 or 1.
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} .