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In an earlier question, I asked about the limiting distribution of the square root of the absolute value of the sum of $n$ i.i.d. random variables each with finite non-zero mean $\mu$ and variance $\sigma^2$. The answer was (after a suitable location and scale adjustment) a standard normal distribution.

This time I am asking about the case where $\mu=0$. I know and can demonstrate the answer, but I am interested in whether the resulting distribution has any other uses.

So suppose $X_1, X_2, \ldots, X_n$ are i.i.d. random variables with zero mean and positive variance $\sigma^2$. If $\displaystyle Z=\sqrt{\left|\sum_{i=1}^n X_i\right|}$ then $\displaystyle \dfrac{Z}{\sqrt[4]{n} \, \sqrt{\sigma}} \ \xrightarrow{d}\ W$ as $n$ increases, where $W$ has the distribution of the square root of a standard half-normal random variable.

  • The density of $W$ is $f(x)=\sqrt{\dfrac{8}{\pi}}\,x\,e^{-x^4/2}$ for $x \ge 0$
  • The mean of $W$ is $\dfrac{\sqrt[4]{2}\;\Gamma\left(\frac34\right)}{\sqrt{\pi}} \approx 0.822179$
  • The variance of $W$ is $\sqrt{\dfrac2\pi}{\left(1- \dfrac{\Gamma\left(\frac34\right)^2}{\sqrt{\pi}}\right)} \approx 0.1219063$
  • The standard deviation of $W$ is then about $0.3491509$
  • So the mean and standard deviation of $Z$ could be approximated by those of $W$ scaled up by a factor of $\sqrt[4]{n} \, \sqrt{\sigma}$

The density of $W$ looks like the following chart. It has a very light tail, with $\mathbb{P}(W>2) < 0.00007$

density square root of a standard half-normal random variable

Has anybody come across this distribution before?

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  • 2
    $\begingroup$ en.wikipedia.org/wiki/Generalized_gamma_distribution $\endgroup$ – whuber Jul 19 '17 at 0:17
  • $\begingroup$ @whuber: Many thanks for that link. So the parameters here for $W$ would be $d=2$ and $p=4$, then scaled with $a=\sqrt[4]{2}$ or for the approximation for $Z$ scaled with $ a=\sqrt[4]{2} \, \sqrt[4]{n} \, \sqrt{\sigma}$ $\endgroup$ – Henry Jul 19 '17 at 1:34
  • $\begingroup$ A standard diagnostic plot for linear regression is to plot the square root of absolute residuals vs the fitted value ("spread-location plot"). The intention may be to make use of the moderately nice shape of this distribution? $\endgroup$ – Paul Harrison Aug 20 '18 at 22:40

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