Random variable

Pure mathematics

Discrete mathematics

Statistics

Materials science

Abelian group

Distribution (mathematics)

Automorphism

Nanotechnology

Mathematics

Characterization (materials science)

Mathematical analysis

Real line

Locally compact space

Theory of Probability and Its Applications

By Heyde's theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of $n$ independent random variables with the other fixed. When $n=2$ we prove analogues of this theorem in the case when independent random variables take values in a locally compact Abelian group $X$ and coefficients of the linear forms are topological automorphisms of $X$.

Heyde's Characterization Theorem for Some Locally Compact Abelian Groups