The arithmetic mean of normal random variables is normal. The geometric mean of log-normal random variables is log-normal. But is there a common distribution family closed under taking harmonic means?
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In general you want to take $X=1/Y$, where $Y$ is a stable random variable. A surprising example is $X=N(0,1)^2$, since $N(0,1)^{-2}$ is stable of index $1/2$. |
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Any class of distributions which is closed under independent sums and almost surely nonzero will work here (and of course will also give an example for geometric means corresponding to the log-normal) the same way as in Michael's example. So besides normal (p=2) and Cauchy (p=1) random variables, the reciprocals of any p-stable random variables work. Of course, only Cauchy have the property of being distributed the same as their reciprocals, so John's answer doesn't generalize this far. |
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It just occurred to me that since Cauchy random variables are closed under reciprocals and closed under sums, they're closed under harmonic means. |
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Perhaps variables of the form 1/X, where X is normal? (We can ignore the problem of division by zero because X is zero only with probability zero.) Of course this isn't exactly common. |
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