The central limit theorem for extremal characters of the infinite symmetric group

The asymptotics of the first rows and columns of random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in $n$, the number of boxes of random diagrams, and prove the central limit theorem for them in the case of distinct Thoma parameters. We also establish a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model. After this paper was completed, the author learned that the central limit theorem has been also proved in the work of M\'eliot (arXiv:1105.0091v1) by a different method.