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

  • $\begingroup$ I flagged this question as off-topic stats.stackexchange.com/help/dont-ask $\endgroup$ – Slow loris May 22 '16 at 1:07
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    $\begingroup$ I think this is on-topic here, it's really a reference request so I have added the tag. $\endgroup$ – Silverfish May 22 '16 at 8:33
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    $\begingroup$ This is clearly on-topic here! probability is a basic part of mathematical statistics! $\endgroup$ – kjetil b halvorsen May 22 '16 at 8:46
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    $\begingroup$ Please note that the various forms of the CLT are about convergence of standardized sample means in distribution. The fact that you mention convergence in probability makes me wonder if you aren't instead asking about convergence like the weak law of large numbers, which is about $\bar{X}_n \xrightarrow{P}\mu$ $\endgroup$ – Glen_b May 22 '16 at 9:01
  • $\begingroup$ Okay, sorry if I screwed up with the flag. I thought asking for paper suggestions was off topic regardless of the desired content of the paper. If the question had been directly asking for help making an argument or proof (which is what Glen_b's answer provides instead of paper recommendations), I wouldn't have flagged. $\endgroup$ – Slow loris May 24 '16 at 11:51

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]


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