  Motivated by a problem in learning theory, we are led to study the dominant
eigenvalue of a class of random matrices. This turns out to be related to the
roots of the derivative of random polynomials (generated by picking their roots
uniformly at random in the interval [0, 1], although our results extend to
other distributions). This, in turn, requires the study of the statistical
behavior of the harmonic mean of random variables as above, and that, in turn,
leads us to delicate question of the rate of convergence to stable laws and
tail estimates for stable laws.
