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One result I deducted was that the ratio of any two of them (eg. X_1 / X_2) is independent of the sample average 1/n * \sum_{i=1}^{n} X_i.

(Aside: that ratio, as a random variable, has a Pareto distribution)

What's the reasoning/ intuitive appeal behind that? I know that any datapoint from an independently, identically distributed sample is in general not independent of the sample average unless there is zero variance, so.. How do we interpret this result here? Why is the ratio independent of the sample average?