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This multiplicative version of the central limit theorem is sometimes called Gibrat's law. Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable. [34]
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 ...
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
Normality: According to the Central Limit Theorem, the distribution of a sum of independent random variables with finite variance tends towards a Gaussian distribution. Loosely speaking, a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original variables.
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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}} .
The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
By definition, a consistent estimator B converges in probability to its true value β, and often a central limit theorem can be applied to obtain asymptotic normality: (,),