Solved – convergence of geometric mean/harmonic mean

Question

Does any one know papers regarding the convergence of geometric mean or harmonic mean in probability, parallel to central limit theorem?

Answer 1

If you're really after something like the weak law of large numbers, note that for positive random variables the log of the geometric mean is arithmetic mean of the logs.

Let $\tilde{X}_n = \left(\prod_{i=1}^n X_i\right)^{1/n}$ for $X_i>0$; and let $Y_i=\log(X_i)$. Then $\bar{Y}_n \xrightarrow{P}\mu_Y$ by the WLLN. Now you can write the $\epsilon_Y$ in terms of the $\epsilon_X$ (or more specifically, if you look at the $\epsilon-\delta$ argument at the link, relate both the $\epsilon$s and the $\delta$s) to demonstrate that there will also be convergence in probability of $\tilde{X}_n$ to $\exp(\mu_Y)$.

A similar argument could be made for the harmonic mean for positive variables.

[I realize you're after references, not arguments, but the arguments appear to be straightforward so you might not easily find a reference outside a textbook exercise]