Characterizations of symmetric distributions based on Rényi entropy

It is proved that the equality of Rényi entropies of upper and lower order statistics as well as upper and lower k-records is a characteristic property of symmetric distributions. Also, for Farlie-Gumbel-Morgenstern (FGM) family, it is shown that under some conditions the equality of entropies of concomitants of upper and lower order statistics as well as concomitants of upper and lower record values is a characteristic property for the uniform distribution.