Law-invariant functionals that collapse to the mean

Invariant Functionals on Speh Representations

We study Sp2n(R)-invariant functionals on the spaces of smooth vectors in Speh representations of GL2n(R). For even n we give expressions for such invariant functionals using an explicit realization of the space of smooth vectors in the Speh representations. Furthermore, we show that the functional we construct is, up to a constant, the unique functional on the Speh representation which is inva...

On Strong Invariant Coordinate System (SICS) Functionals

In modern multivariate statistical analysis, affine invariance or equivariance for statistical procedures are properties of paramount interest and importance. A statistical procedure lacking such a property can sometimes acquire it if carried out not on the original data but rather on suitably transformed data, in some cases accompanied by a retransformation back to the original coordinate syst...

On Invariant Coordinate System (ICS) Functionals

Equivariance and invariance issues often arise in multivariate statistical analysis. Statistical procedures have to be modified sometimes to obtain an affine equivariant or invariant version. This is often done by preprocessing the data, e.g., by standardizing the multivariate data or by transforming the data to an invariant coordinate system. In this article standardization of multivariate dis...

Conservation laws for invariant functionals containing compositions

The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized EulerLagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work we prove a generalization of the necessary optimality condition of DuBois-Rey...

Topologically Left Invariant Mean on Dual Semigroup Algebras