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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
. As an example of very slow convergence,
let us imagine independent variables described by a Poisson
distribution of
: 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!

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Giulio D'Agostini
2003-05-15