Convergence of Percentile in Power Law I have a probability distribution, that in its tail follows a power law. I've noticed, while I was simulating samples, and determining parameters experimentally, that as I increase the value of a percentile I want to measure experimentally, the percentile converges ever so slowly. For instance the median is approximated within 2% after 100 samples, the 75% percentile requires about 500 samples, and the 95% percentile requires several thousand samples. I imagine there is a way to determine the distribution of the percentile error, and I was trying to use the methods used by Newman (2005) to derive a formula, but I'm not really getting anywhere on my own. Are there any?
Reference
Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf’s law. Contemporary Physics, 46(5), 323–351. https://doi.org/10.1080/00107510500052444
 A: After several comments linking to detailed explanations for a separate question, here is a quick summary.
Any event that has a probability p of happening, follows a binomial distribution, and so do percentiles or quantiles. The variance of a binomial is $n p (1-p)$ for the number of data points included on either side of the percentile, therefore the standard deviation is $\sigma = \sqrt{ n p(1-p)}\ $. If we want the standard deviation as proportion, it is $\sigma = \sqrt{ p(1-p)/n}\ $. Because this proportion represents an area of variation under the probability density function PDF, and because the PDF calculated at percentile p, $PDF(p)$, represents the height of that area, we can see that the width of that area will be $\sigma /PDF(p) $, which is the standard deviation of the position of the percentile.
As noted in one of the answers linked above, for extreme values of percentiles, the binomial distribution departs significantly from a normal distribution; but keeping that in mind it will not be too difficult to calculate asymmetric confidence intervals.
In particular for the distribution I was dealing with (Cauchy), which has a left tail asymptotic to a power law (exponent -1), to obtain approximately the same standard deviation of 0.09, for the median $=0\pm0.09$, 300 points are required, for the 75% percentile $=1\pm0.09$, 900 points are required, and for the 95% percentile$=6.3\pm0.09$, 90,000 points are required.
I would like to thank Whuber for leading me through the path to get to the right answer.
