On a characterisation theorem for $a$-adic solenoids

TL;DR

This paper extends Heyde's characterization theorem from real numbers to $a$-adic solenoids, identifying conditions under which the distribution of random variables is uniquely determined by the symmetry of a conditional distribution.

Contribution

It proves an analogue of Heyde's theorem for $a$-adic solenoids, broadening the understanding of distribution characterization in topological groups.

Findings

Characterization holds for $a$-adic solenoids without elements of order 2.

Conditions include non-vanishing characteristic functions and automorphic coefficients.

The theorem generalizes classical results to a broader class of topological groups.

Abstract

According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We prove an analogue of this theorem for linear forms of two independent random variables taking values in an $a$-adic solenoid $\Sigma_a$ without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of $\Sigma_a$.