On a characterization theorem in the space \(\mathbb{R}^n\) (Q6177091)

The author establishes an analogue in \(\mathbb{R}^n\) of Heyde's theorem characterizing Gaussian distributions on the real line by the symmetry of one linear form of independent random variables conditioned on another. In particular, let \(\alpha\) be an invertible linear operator in \(\mathbb{R}^n\), and \(\xi_1\) and \(\xi_2\) independent random vectors. The author shows that if the distribution of \(\xi_1+\alpha\xi_2\) conditional on \(\xi_1+\xi_2\) is symmetric, then the distributions of the \(\xi_j\) are shifts of convolutions of symmetric Gaussian distributions supported on an \(\alpha\)-invariant subspace \(G\), and a distribution supported on \(K=\text{Ker}(I+\alpha)\), where \(I\) is the identity. It also holds that \(G\cap K=\{0\}\). This result is of principal interest in the case where \(I+\alpha\) is not invertible.