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I want to know how Lindeberg came up with the condition which is sufficient for CLT to hold ? What is the intuition behind such an expression ?

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This is what Lindeberg says himself about his inspiration:

J.W. Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung (1922). [I translate from the German]

In my 1920 paper "On the exponential law in probability theory" I proved several theorems, related to the question under which conditions the sum of a large number of independent stochastic variables follows the Gaussian distribution. At that time of writing I held the theorem of Von Mises (1919) as the sharpest result for this question. However, now I have found that previously (1900) Lyapunov derived some general results, which not only go beyond those of Von Mises, but from which moreover most of the theorems from my paper can be derived. [...] This has motivated me to return to my previous work, and I have found that my method can be simplified considerably, and that a small modification leads to a substantial improvement of all known earlier results.

So it would seem that Lindeberg got his inspiration from Lyapunov.

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I wanted to the technical aspect, not this kind of motivation. how the thought of the expression in lindeberg condition ? – aaaaaa Apr 4 '14 at 18:01
you asked how Lindeberg came up with his condition --- he read Lyapunov, understood his paper and realized he could weaken the condition (as you can see, Linderberg and Lyapunov conditions are quite similar). I think that's how it went, don't you think? – Carlo Beenakker Apr 4 '14 at 20:47

It was Lyapunov who introduced the method of characteristic functions and used it to derive CLT under certain conditions on moments of the random variables involved. His moment conditions were sufficient but not necessary for CLT. Lindeberg conditions are weakened Lyapunov conditions. Their advantage is that under additional assumptions they are necessary and sufficient for CLT. Lindeberg's conditions are tightly connected to "uniform smallness": the contribution of one concrete r.v. into the sum is small compared to the entire sum.

A classical book on i.i.d. summation is by Gnedenko & Kolmogorov. A more recent, almost exhaustive and self-contained monograph on classical summation theory is by V.V.Petrov.

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