Since the events that the i'th entry is a record are independent, doesn't this follow from a central limit like theorem? The number of records is a sum of independent binomial random variables, Bin(1,1/i). Cris Moore
--actually (after correction) this idea does work. The kth random variable (1 if new record, 0 if not) out of N has mean=1/k and variance=1/k-1/k^2. This instantly proves that mean=harmonic(N) and var=H(N,1)-H(N,2), the simplest such proof. Further, although the Lyapunov CLT I was using does not work in this scenario to prove normality, the Lindeberg-Feller CLT after addititive corrections the make the mean=0, *does* work. (Also described by John D. Cook: http://www.johndcook.com/central_limit_theorems.html). And that is the simplest proof of normality too (given these CLTs available) although somewhat deceptive since my old Lyapunov-based proof (which apparently was what Renyi had used too) makes the slow convergence clear, while this does not.