Mathematisches Kolloquium

Zusammenfassung: The central limit theorem is one of the most famous results in probability theory and it has been the source of many research works. Besides probability theory, central limit theorems are established in mathematical physics (e.g. in random matrix theory), in number theory (the Selberg central limit theorem for the Riemann zeta function on the critical line or the Erdös-Kac theorem for the total number of distinct divisors of an integer n), in combinatorics, etc. Roughly speaking, the central limit theorem consists in showing that, when properly normalized, the sequence of random variables under consideration converges in distribution. In this talk, we shall see that one can sometimes obtain more rened results by seeking for a convergence after renormalization of the Fourier transform of the laws of the random variables (rather than the random variables themselves). This will lead us to introduce a new mode of convergence which we shall illustrate with examples mainly from random matrix theory and number theory.

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