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
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