By the Strong Law of Large Numbers applied to log(X) the geometric mean of X_1,...,X_n > 0 and IID like X converges toexp(E[log(X)]] which, by Jensen's inequality, is always? <= E[X] and is strictly less than E[X] except in trivial extreme cases. In short: by using the geometric mean all asymptotic results no longer apply. Michael Meyer [[alternative HTML version deleted]]