A Characterization Based on the Absolute Difference of Two I. I. D. Random Variables
Prem Puri, Herman Rubin · The Annals of Mathematical Statistics · 1970
Let X be a nonnegative random variable with X sub 1 and X sub 2 as its two independent copies. The problem considered here is to characterize all the nonnegative distributions with the property that the distribution of the absolute difference /(X sub 1)-(X sub 2)/ is the same as that of X. It is shown that in general such a distribution has to be either purely discrete, or absolute continuous or singular and that it cannot be their mixture.