  We study the convergence properties of a pair of learning algorithms
(learning with and without memory). This leads us 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, which leads us to
delicate question of the rate of convergence to stable laws and tail estimates
for stable laws. The reader can find the proofs of most of the results
announced here in the paper entitled "Harmonic mean, random polynomials, and
random matrices", by the same authors.
