An extremal property of the generalized arcsine distribution

The main result of the paper is the following characterization of the generalized arcsine density p γ (t) = t γ−1 (1 − t) γ−1 /B(γ, γ) with $${t \in (0, 1)}$$ and $${\gamma \in(0,\frac12) \cup (\frac12,1)}$$ : a r.v. ξ supported on [0, 1] has the generalized arcsine density p γ (t) if and only if $${ {\mathbb E} |\xi- x|^{1-2 \gamma}}$$ has the same value for almost all $${x \in (0,1)}$$ . Moreover, the measure with density p γ (t) is a unique minimizer (in the space of all probability measures μ supported on (0, 1)) of the double expectation $${ (\gamma-\frac12 ) {\mathbb E} |\xi-\xi^{\prime}|^{1-2 \gamma}}$$ , where ξ and ξ′ are independent random variables distributed according to the measure μ. These results extend recent results characterizing the standard arcsine density (the case $${\gamma=\frac12}$$ ). Copyright Springer-Verlag 2013