Solved – When to use Central Limit Theorem

  1. Is the central-limit-theorem true for all distributions?
  2. For what sample size is it true (what is meant with "large")?

There are many versions of the Central Limit Theorem. A common one applies to distributions which have finite mean and variance. Other versions apply to vectors, sums which are only close to independent, sums of random variables which are not identically distributed, etc.

The conclusion of the Central Limit Theorem doesn't make sense for distributions which don't have finite mean and variance. The average of $n$ independent draws from a standard Cauchy distribution is not approximately normally distributed, it has a standard Cauchy distribution again.

The number of draws needed before the sum or average is close to a normal distribution (in the sense of the maximum discrepancy of the cumulative distribution functions) depends on the distribution. For any $n$, you can choose a distribution whose sum still doesn't look close to normal after $n$ draws. However, there are improvements on the Central Limit Theorem such as the Berry-Esseen Theorem which give you bounds on how many draws you need for a given level of accuracy in terms of things like the normalized absolute third moment $E((X-mu(X))^3)/sigma^3$.

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