On Convolutions and Linear Combinations of Pseudo-Isotropic Distributions

Let E be a Banach space, and let E* be its dual. For understanding the main results of this paper it is enough to consider E= $$\mathbb{R}$$ n . A symmetric random vector X taking values in E is called pseudo-isotropic if all its one-dimensional projections have identical distributions up to a scale parameter, i.e., for every ξ∈E* there exists a positive constant c(ξ) such that (ξ, X) has the same distribution as c(ξ) X 0, where X 0 is a fixed nondegenerate symmetric random variable. The function c defines a quasi-norm on E*. Symmetric Gaussian random vectors and symmetric stable random vectors are the best known examples of pseudo-isotropic vectors. Another well known example is a family of elliptically contoured vectors which are defined as pseudo-isotropic with the quasi-norm c being a norm given by an inner product on E*. We show that if X and Y are independent, pseudo-isotropic and such that X+Y is also pseudo-isotropic, then either X and Y are both symmetric α-stable, for some α∈(0, 2], or they define the same quasi-norm c on E*. The result seems to be especially natural when restricted to elliptically contoured random vectors, namely: if X and Y are symmetric, elliptically contoured and such that X+Y is also elliptically contoured, then either X and Y are both symmetric Gaussian, or their densities have the same level curves. However, even in this simpler form, this theorem has not been proven earlier. Our proof is based upon investigation of the following functional equation: $$\begin{gathered} \exists m \in \left( {0,1} \right)\forall a,b >0,{a \mathord{\left/ {\vphantom {a {b \in \left[ {m,m^{ - 1} } \right]}}} \right. \kern \nulldelimiterspace} {b \in \left[ {m,m^{ - 1} } \right]}}\exists q\left( {a,b} \right) > 0\forall t > 0 \hfill \\\varphi \left( {at} \right)\psi \left( {bt} \right) = {\chi }\left( {q\left( {a,b} \right)t} \right) \hfill \\ \end{gathered}$$ which we solve in the class of real characteristic functions.