An extremal property of the generalized arcsine distribution

The main result of the paper is the following characterization of the generalized arcsine density p <Subscript> γ </Subscript>(t) = t <Superscript> γ−1</Superscript>(1 − t)<Superscript> γ−1</Superscript>/B(γ, γ)   with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${t \in (0, 1)}$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\gamma \in(0,\frac12) \cup (\frac12,1)}$$</EquationSource> </InlineEquation> : a r.v. ξ supported on [0, 1] has the generalized arcsine density p <Subscript> γ </Subscript>(t) if and only if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${ {\mathbb E} |\xi- x|^{1-2 \gamma}}$$</EquationSource> </InlineEquation> has the same value for almost all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${x \in (0,1)}$$</EquationSource> </InlineEquation> . Moreover, the measure with density p <Subscript> γ </Subscript>(t) is a unique minimizer (in the space of all probability measures μ supported on (0, 1)) of the double expectation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${ (\gamma-\frac12 ) {\mathbb E} |\xi-\xi^{\prime}|^{1-2 \gamma}}$$</EquationSource> </InlineEquation> , 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${\gamma=\frac12}$$</EquationSource> </InlineEquation>). Copyright Springer-Verlag 2013