Harmonic mean, random polynomials and stochastic matrices

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. Introduction The original motivation for the work in this paper was provided by the first-named author’s research in learning theory, specifically in various models of language acquisition (see [KNN2001, NKN2001, KN2001]) and more specifically yet by the analysis of the speed of convergence of the memoryless learner algorithm. The setup is described in some detail in section 3.1, but here we will just recall the essentials: there is a collection of concepts R1, . . . , Rn and words which refer to this concepts, sometimes ambiguously. The teacher generates a stream of words, referring to the concept R1. This is not known to the student, but he must learn by, at each steps, guessing some concept Ri and checking for consistency with the teacher’s input. The memoryless learner algorithm consists of picking a concept Ri at random, and sticking by this choice, until it is proven wrong. At this point another concept is picked randomly, and the procedure repeats. It is clear that once the student hits on the right answer R1, this will be his final answer, so the question is then: 1991 Mathematics Subject Classification. 60E07, 60F15, 60J20, 91E40, 26C10.