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

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#### Best Answer

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 $tilde{X}_n = left(prod_{i=1}^n X_iright)^{1/n}$ 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]