On a Characterization Theorem for Locally Compact Abelian Groups Containing an Element of Order 2
- PDF / 455,626 Bytes
- 19 Pages / 439.642 x 666.49 pts Page_size
- 54 Downloads / 208 Views
On a Characterization Theorem for Locally Compact Abelian Groups Containing an Element of Order 2 G. M. Feldman1 Received: 8 April 2020 / Accepted: 4 November 2020 / © Springer Nature B.V. 2020
Abstract According to the well-known 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 the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. While coefficients of linear forms are topological automorphisms of a group. Keywords Characterization theorem · Conditional distribution · Topological automorphism · Locally compact Abelian group Mathematics Subject Classification (2010) 60B15 · 62E10 · 43A35
1 Introduction In [12], see also [14, §13.4.1], C.C. Heyde proved the following characterization theorem for the Gaussian distribution on the real line.
The Heyde theorem Let ξj , j = 1, 2, . . . , n, n ≥ 2, be independent random variables
with distributions μj . Let aj , bj be nonzero real numbers satisfying the conditions bi ai−1 + bj aj−1 = 0 for all i, j . Assume that the conditional distribution of the linear form L2 = b1 ξ1 + · · · + bn ξn given L1 = a1 ξ1 + · · · + an ξn is symmetric. Then all distributions μj are Gaussian, possibly degenerate. Some analogues of Heyde’s theorem in the case when independent random variables take values in a locally compact Abelian group X, and coefficients of linear forms are topological automorphisms of X were studied in [2–4, 6–11, 15–17], see also [5, Chapter VI]. In this G. M. Feldman
[email protected] 1
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47, Nauky Ave, Kharkiv, 61103, Ukraine
G.M. Feldman
article we continue to study group analogues of the Heyde theorem. We consider groups that contain an element of order 2. Elements of order 2 play a special role in the Heyde theorem. As shown in the article, if a group X contains an element of order 2, it leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. Let X be a second countable locally compact Abelian group. Denote by Aut(X) the group of topological automorphisms of X, and by I the identity automorphism of a group. Denote by Y the character group of the group X, and by (x, y) the value of a character y ∈ Y at an element x ∈ X. If K is a closed subgroup of X, denote by A(Y, K) = {y ∈ Y : (x, y) = 1 for all x ∈ K} its annihilator. Let X1 and X2 be locally compact Abelian groups with the character groups Y1 and Y2 respectively. Let α : X1 → X2 be a continuous
Data Loading...
