Caution

Following this commercial in favour of the miraculous properties of the central limit theorem, some words of caution are in order:

• Although I have tried to convince the reader that the convergence is rather fast in the cases of practical interest, the theorem only states that the asymptotic Gaussian distribution is reached for $n\rightarrow\infty$. As an example of very slow convergence, let us imagine $10^9$ independent variables described by a Poisson distribution of $\lambda_i=10^{-9}$: their sum is still far from a Gaussian.
• Sometimes the conditions of the theorem are not satisfied.
  • A single component dominates the fluctuation of the sum: a typical case is the well-known Landau distribution; systematic errors may also have the same effect on the global error.
  • The condition of independence is lost if systematic errors affect a set of measurements, or if there is coherent noise.
• The tails of the distributions do exist and they are not always Gaussian! Moreover, realizations of a random variable several standard deviations away from the mean are possible. And they show up without notice!