On a group analogue of the Heyde theorem

TL;DR

This paper extends Heyde's theorem, which characterizes Gaussian distributions via symmetry conditions, to the setting of locally compact Abelian groups with integer coefficients in linear forms.

Contribution

It introduces a group-theoretic analogue of Heyde's theorem, broadening the understanding of distribution characterization in more general algebraic structures.

Findings

Established a characterization of Gaussian distributions on locally compact Abelian groups.

Demonstrated the conditions under which symmetry of conditional distributions implies Gaussianity.

Extended classical results to a more general algebraic setting.

Abstract

Heyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analog of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.

Equations

$$\hat{\gamma }(y)=(x,y)\exp \{-\phi (y)\},$$

$$\phi (u+v)+\phi (u-v)=2[\phi (u)+\phi (v)],u,v\in Y.$$

$$\widehat{m}_{K}(y)=\left\{\begin{array}[]{ll}1,&\hbox{$y\in A(Y,K)$;}\\ 0,&\hbox{$y\not\in A(Y,K)$.}\\ \end{array}\right.$$

$${\Delta }_{h}f(y)=f(y+h)-f(y).$$

$${\Delta }_h^{n+1}f(y)=0$$

$$\mathbb{Z}(p{)}^{\mathfrak{m}}\times \mathbb{Z}(p{)}^{\mathfrak{n}\ast },$$

$$\prod _{j=1}^n{\hat{\mu }}_j(a_{j}u+b_{j}v)=\prod _{j=1}^n{\hat{\mu }}_j(a_{j}u-b_{j}v),u,v\in Y.$$

$$\mathrm{P}\{\omega \in \Omega :L_1(\omega )\in A,L_2(\omega )\in B\}=\mathrm{P}\{\omega \in \Omega :L_1(\omega )\in A,L_2(\omega )\in -B\}$$

$$\mathbf{E}\left[(L_1,u)(L_2,v)\right]=\mathbf{E}\left[(L_1,u)(-L_2,v)\right],u,v\in Y.$$

$$\mathbf{E}\left[(a_1{\xi }_1+\cdots +a_n,u)(b_1{\xi }_1+\cdots +b_n{\xi }_n,v)\right]=$$

$$=\mathbf{E}\left[(a_1{\xi }_1+\cdots +a_n,u)(-(b_1{\xi }_1+\cdots +b_n{\xi }_n){\xi }_j,v)\right],u,v\in Y,⟺$$

$$\mathbf{E}\left[\prod _{j=1}^n(a_j{\xi }_j,u)\prod _{j=1}^n(b_j{\xi }_j,v)\right]=\mathbf{E}\left[\prod _{j=1}^n(a_j{\xi }_j,u)\prod _{j=1}^n(-b_j{\xi }_j,v)\right],u,v\in Y,⟺$$

$$\mathbf{E}\left[\prod _{j=1}^n({\xi }_j,a_{j}u)\prod _{j=1}^n({\xi }_j,b_{j}v)\right]=\mathbf{E}\left[\prod _{j=1}^n({\xi }_j,a_{j}u)\prod _{j=1}^n({\xi }_j,-b_{j}v)\right],u,v\in Y,⟺$$

$$\mathbf{E}\left[\prod _{j=1}^n({\xi }_j,a_{j}u+b_{j}v)\right]=\mathbf{E}\left[\prod _{j=1}^n({\xi }_j,a_{j}u-b_{j}v)\right],u,v\in Y.$$

$$\prod _{j=1}^n\mathbf{E}[({\xi }_j,a_{j}u+b_{j}v)]=\prod _{j=1}^n\mathbf{E}[({\xi }_j,a_{j}u-b_{j}v)],u,v\in Y.$$

$$\hat{\mu }(y)=\exp \{\psi (y)\},\psi (0)=0,y\in Y,$$

$$\sum _{j=1}^n{\psi }_j(a_{j}u+b_{j}v)=\sum _{j=1}^n{\psi }_j(a_{j}u-b_{j}v),u,v\in Y.$$

$$\sum _{j=1}^n{\Delta }_{l_{nj}{h}_n}{\psi }_j(a_{j}u+b_{j}v)=\sum _{j=1}^{n-1}{\Delta }_{m_{nj}{h}_n}{\psi }_j(a_{j}u-b_{j}v),u,v\in Y,$$

$$\sum _{j=1}^n{\Delta }_{l_{n-1,j}{h}_{n-1}}{\Delta }_{l_{nj}{h}_n}{\psi }_j(a_{j}u+b_{j}v)=\sum _{j=1}^{n-2}{\Delta }_{m_{n-1,j}{h}_{n-1}}{\Delta }_{m_{nj}{h}_n}{\psi }_j(a_{j}u-b_{j}v),u,v\in Y,$$

$$\sum _{j=1}^n{\Delta }_{l_{1j}{h}_1}{\Delta }_{l_{2j}{h}_2}...{\Delta }_{l_{nj}{h}_n}{\psi }_j(a_{j}u+b_{j}v)=0,u,v\in Y,$$

$$\sum _{j=1}^{n-1}{\Delta }_{m_{nj}{k}_n}{\Delta }_{l_{1j}{h}_1}{\Delta }_{l_{2j}{h}_2}...{\Delta }_{l_{nj}{h}_n}{\psi }_j(a_{j}u+b_{j}v)=0,u,v\in Y.$$

$${\Delta }_{m_{21}{k}_2}...{\Delta }_{m_{n1}{k}_n}{\Delta }_{l_{11}{h}_1}{\Delta }_{l_{2,1}{h}_2}...{\Delta }_{l_{n1}{h}_n}{\psi }_1(a_{1}u+b_{1}v)=0,u,v\in Y.$$

$${\Delta }_{m_{21}{k}_2}...{\Delta }_{m_{n1}{k}_n}{\Delta }_{l_{11}{h}_1}{\Delta }_{l_{2,1}{h}_2}...{\Delta }_{l_{n1}{h}_n}{\psi }_1(a_{1}u)=0,u\in Y.$$

$${\Delta }_h^{2n-1}{\psi }_1(y)=0,y,h\in Y.$$

$$\frac{a_1}{b_1}=\cdots =\frac{a_{r_1}}{b_{r_1}}=c_1,\frac{a_{r_1+1}}{b_{r_1+1}}=\cdots =\frac{a_{r_2}}{b_{r_2}}=c_2,\dots ,\frac{a_{r_k+1}}{b_{r_k+1}}=\cdots =\frac{a_n}{b_n}=c_{k+1},$$

$$d_0=\frac{b_1{b}_{r_1+1}\dots b_{r_k+1}}{b_1},d_j=\frac{b_1{b}_{r_1+1}\dots b_{r_k+1}}{b_{r_j+1}},j=1,2,\dots ,k,$$

$${\zeta }_1=b_1{\xi }_1+\cdots +b_{r_1}{\xi }_{r_1},{\zeta }_2=b_{r_1+1}{\xi }_{r_1+1}+\cdots +b_{r_2}{\xi }_{r_2},\dots ,{\zeta }_{k+1}=b_{r_k+1}{\xi }_{r_k+1}+\cdots +b_n{\xi }_n.$$

$$L_1^{\prime }=a_1{d}_0{\zeta }_1+a_{r_1+1}{d}_1{\zeta }_2+\cdots +a_{r_k+1}{d}_k{\zeta }_{k+1},L_2^{\prime }={\zeta }_1+{\zeta }_2+\cdots +{\zeta }_{k+1}.$$

$$\mu =a{E}_0+(1-a)m_M,$$

$$\widehat{\mu}(y)=\left\{\begin{array}[]{ll}1,&\hbox{$y\in A(Y,M)$,}\\ a,&\hbox{$y\not\in A(Y,M)$.}\\ \end{array}\right.$$

Full text

Key words and phrases: locally compact Abelian group, Gaussian distribution, Haar distribution, Heyde theorem, independence, Q-independence

2010 Mathematics Subject Classification: Primary 60B15; Secondary 62E10.

1 Introduction

In 1953 V.P.Skitovich and G.Darmois proved independently that a Gaussian distribution on a real line is characterized by the independence of two linear forms of $n$ independent random variables ([12, $\S$3.1]). In 1970 C.C.Heyde proved a similar result where a Gaussian distribution is characterized by the symmetry of the conditional distribution of one linear form given another.

The Heyde theorem ([11]). Let $\xi_{j},\ j=1,2,...,n,\ n\geq 2,$ be independent random variables. Consider linear forms $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ and $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$, where the coefficients $a_{j},b_{j}$ are nonzero real numbers such that $b_{i}a_{j}+b_{j}a_{i}\neq 0$ for all $i,j$. If the conditional distribution of $L_{2}$ given $L_{1}$ is symmetric then all random variables $\xi_{j}$ are Gaussian.

The Skitovich-Darmois theorem and the Heyde theorem were proved using the finite difference method.

The Skitovich-Darmois theorem and the Heyde theorem were generalized on locally compact Abelian groups (see e.g. [4], [5], [8], [14], [15], [16]). Specifically, in the article [3] G.Feldman proved a generalization of the Skitovich-Darmois theorem in the case when random variables take values in a locally compact Abelian group and coefficients of the linear forms are integers. The main result of this article is a generalization of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and coefficients of the linear forms are integers (see §3). To obtain this result we use methods, which differ from methods of the article [3].

In the article [13] A.M.Kagan and G.J.Székely introduced a notion of $Q$-independence of random variables which generalizes a notion of independence of random variables. In particular, they proved that some classical characterization theorems of mathematical statistics hold true if instead of independence $Q$-independence is considered. The article [13] has stimulated a series of studies. Generalizations of some characterization theorems on locally compact Abelian groups for $Q$-independent random variables were obtained in [7], [17], [18]. In §4 we prove that the results of §3 hold true if instead of independence $Q$-independence is considered.

To prove the main results of this article, we use the finite-difference-method.

2 Notation and definitions

In the article we use standard results on abstract harmonic analysis (see e.g. [9]).

Let $X$ be a second countable locally compact Abelian group, $Y=X^{\ast}$ be its character group, and $(x,y)$ be the value of a character $y\in Y$ at an element $x\in X$. Let $H$ be a subgroup of $Y$. Denote by $A(X,H)=\{x\in X:(x,y)=1\ \ \forall\ y\in H\}$ the annihilator of $H$. For each integer $n$, $n\neq 0,$ let $f_{n}:X\mapsto X$ be the endomorphism $f_{n}x=nx.$ Set $X^{(n)}=f_{n}(X)$, $X_{(n)}=Kerf_{n}$. Denote by ${\mathbb{Z}}(n)$ the finite cyclic group of order $n$. For a fixed prime $p$ denote by ${\mathbb{Z}}(p^{\infty})$ the set of rational numbers of the form $\{k/p^{n}:k=0,1,...,p^{n}-1,n=0,1,...\}$ and define the operation in ${\mathbb{Z}}(p^{\infty})$ as addition modulo 1. Then ${\mathbb{Z}}(p^{\infty})$ is transformed into an Abelian group, which we consider in the discrete topology. Denote by $\Delta_{p}$ the group of $p$-adic integers ([9, §10.2]). Note that $\Delta_{p}^{*}\approx{\mathbb{Z}}(p^{\infty})$ ([9, §25.2]).

Let ${M^{1}}(X)$ be the convolution semigroup of probability distributions on $X$, $\widehat{\mu}(y)=\int_{X}(x,y)d\mu(x)$ be the characteristic function of a distribution $\mu\in{M^{1}}(X)$, and $\sigma(\mu)$ be the support of $\mu$. If $H$ is a closed subgroup of $Y$ and $\widehat{\mu}(y)=1$ for $y\in H$, then $\widehat{\mu}(y+h)=\widehat{\mu}(y)$ for all $y\in Y,\ h\in H$ and $\sigma(\mu)\subset A(X,H)$. For $\mu\in{M^{1}}(X)$ we define the distribution $\bar{\mu}\in M^{1}(X)$ by the rule $\bar{\mu}(B)=\mu(-B)$ for all Borel sets $B\subset X$. Observe that $\widehat{\bar{\mu}}(y)=\overline{\widehat{\mu}(y)}$.

Let $x\in X$. Denote by $E_{x}$ the degenerate distribution concentrated at the point $x$, and by $D(X)$ the set of all degenerate distributions on $X$. A distribution $\gamma\in{M^{1}}(X)$ is called Gaussian ([19, §4.6]) if its characteristic function can be represented in the form

[TABLE]

where $x\in X$ and $\varphi(y)$ is a continuous nonnegative function satisfying the equation

[TABLE]

Denote by $\Gamma(X)$ the set of Gaussian distributions on $X$. We note that according to this definition $D(X)\subset\Gamma(X)$. Denote by $I(X)$ the set of shifts of Haar distributions $m_{K}$ of compact subgroups $K$ of the group $X$. Note that

[TABLE]

We note that if a distribution $\mu\in\Gamma(X)*I(X)$, i.e. $\mu=\gamma*m_{K}$, where $\gamma\in\Gamma(X)$, then $\mu$ is invariant with respect to the compact subgroup $K\subset X$ and under the natural homomorphism $X\mapsto X/K\ \mu$ induces a Gaussian distribution on the factor group $X/K$. Therefore the class $\Gamma(X)*I(X)$ can be considered as a natural analogue of the class $\Gamma(X)$ on locally compact Abelian groups.

Let $f(y)$ be a function on $Y$, and $h\in Y.$ Denote by $\Delta_{h}$ the finite difference operator

[TABLE]

A function $f(y)$ on $Y$ is called a polynomial if

[TABLE]

for some $n$ and for all $y,h\in Y$.

An integer $a$ is said to be admissible for a group $X$ if $X^{(a)}\neq\{0\}$. The admissibility of integers $a_{1},\dots,a_{n}$ when we consider the linear form $a_{1}\xi_{1}+\dots+a_{n}\xi_{n}$ is a group analogue of the condition $a_{j}\neq 0,\ j=1,2,\dots,n,$ for the case of $X=\mathbb{R}$.

3 The Heyde theorem for locally compact Abelian groups

The main result of this section is a full description of locally compact Abelian groups for which the group analogue of the Heyde theorem takes place in the case when the characteristic functions of considering random variables do not vanish.

Theorem 1. Let $X$ be a second countable locally compact Abelian group such that $X\neq X_{(2)}$. Then the following statements hold:

(I) Let $\xi_{j},\ j=1,2,...,n,\ n\geq 2,$ be independent random variables with values in $X$ and distributions $\mu_{j}$ with non-vanishing characteristic functions. Consider the linear forms $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ and $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$, where the coefficients $a_{j},\ b_{j}$ are admissible integers for $X$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$. Assume that the conditional distribution of $L_{2}$ given $L_{1}$ is symmetric. Then the following statements hold:

$(i)$* If $X$ is a torsion-free group then all $\mu_{j}\in\Gamma(X)$;*

$(ii)$* If $X=X_{(p)}$, where $p$ is a prime number ($p>2$), then all $\mu_{j}\in D(X)$.*

(II) If $X$ is not isomorphic to any of the groups mentioned in (I), then there exist independent random variables $\xi_{j},j=1,2,...,n,n\geq 2,$ with values in $X$ and distributions $\mu_{j}$ with non-vanishing characteristic functions, and admissible integers $a_{j},\ b_{j}$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$, such that the conditional distribution of $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$ given $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ is symmetric, but all $\mu_{j}\not\in\Gamma(X)$.

Remark 1. Let $X$ be a locally compact Abelian group such that every element of $X$ different from zero has order $p$, where $p$ is a fixed prime number. Then $X$ is topologically isomorphic to the group

[TABLE]

where $\mathfrak{m}$ and $\mathfrak{n}$ are arbitrary cardinal numbers, ${\mathbb{Z}(p)}^{\mathfrak{m}}$ is considered in the product topology, and ${\mathbb{Z}(p)}^{\mathfrak{n}*}$ is considered in the discrete topology ([9, §25.29]). If a group $X$ is of the form (3) for $p=2$ then there are no exist admissible integers $a_{j},\ b_{j}$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for all $i,j$ for a group $X$. If a group $X$ is of the form (3) for $p=3$ then the set of admissible integers $a_{j},\ b_{j}$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for all $i,j$ for a group $X$ is either $a_{j}=b_{j}=1$ for all $j$ or $a_{j}=b_{j}=2$ for all $j$.

To prove Theorem 1 we need some lemmas.

The following lemma was proved earlier for the case when the coefficients of linear forms are topological automorphisms of the group ([6, §16.1]). In the case when the coefficients are integers, the proof is the same. For completeness of the presentation, we give it in the article.

Lemma 1. Let $\xi_{j},\ j=1,2,...,n,\ n\geq 2,$ be independent random variables with values in a second countable locally compact Abelian group $X$ and with distributions $\mu_{j}$. Let $a_{j},\ b_{j}$ be integers. The conditional distribution of the linear form $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$ given $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ is symmetric if and only if the characteristic functions $\widehat{\mu}_{j}(y)$ satisfy the equation

[TABLE]

Proof. Let $(\Omega,\mathfrak{A},\mathrm{P})$ be a probabilistic space, where the random variables $\xi_{j}$ are defined. The condition of the symmetry of the conditional distribution of $L_{2}$ given $L_{1}$ is equivalent to the equality

[TABLE]

for all Borel sets $A,B\subset X$. This means that the random variables $(L_{1},L_{2})$ and $(L_{1},-L_{2})$ are identically distributed. It is equivalent to the equality

[TABLE]

We obtain from the form of the linear forms that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Taking into account the independence of the random variables $\xi_{j}$, we have

[TABLE]

Taking into account that $\widehat{\mu}_{j}(y)=\mathbf{E}[(\xi_{j},y)]$, we obtain (4). $\blacksquare$

The following lemma is an analogue of the Cramer theorem on the decomposition of a Gaussian distribution on locally compact Abelian groups.

Lemma 2 ([1], see also [6, §4.6]). Let $X$ be a second countable locally compact Abelian group. Let $\mu=\mu_{1}*\mu_{2}$, where $\mu\in\Gamma(X)$, $\mu_{1},\mu_{2}\in M^{1}(X)$. If $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$ then $\mu_{1},\mu_{2}\in\Gamma(X)$.

The following lemma is an analogue of the Marcinkiewicz theorem on locally compact Abelian groups.

Lemma 3 ([2], see also [6, §5.11]). Let $X$ be a second countable locally compact Abelian group. Let $\mu\in M^{1}(X)$ and its characteristic function is of the form

[TABLE]

where $\psi(y)$ is a continuous polynomial. If $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$ then $\mu\in\Gamma(X)$.

Proof of Theorem 1. (I) It follows from Lemma 1 that the characteristic functions $\widehat{\mu}_{j}(y)$ satisfy equation (4).

Put $\nu_{j}=\mu_{j}*\bar{\mu}_{j}$. Then $\widehat{\nu}_{j}(y)=|\widehat{\mu}_{j}(y)|^{2}>0$. Obviously, the characteristic functions $\widehat{\nu}_{j}(y)$ also satisfy equation (4). If we prove that $\nu_{j}\in\Gamma(X)$, then applying Lemma 2, we obtain that $\mu_{j}\in\Gamma(X)$. Hence we can assume from the beginning that all $\widehat{\mu}_{j}(y)>0$.

Set $\psi_{j}(y)=-\log\widehat{\mu}_{j}(y)$. We conclude from (4) that

[TABLE]

We use the finite difference method to solve equation (7).

Let $h_{n}$ be an arbitrary element of the group $Y$. Substitute $u+b_{n}h_{n}$ for $u$ and $v+a_{n}h_{n}$ for $v$ in equation (7). Subtracting equation (7) from the resulting equation we obtain

[TABLE]

where $l_{nj}=a_{j}b_{n}+b_{j}a_{n}$, $m_{nj}=a_{j}b_{n}-b_{j}a_{n}$. Note that the right-hand side of equation (8) does not contain the function $\psi_{n}$.

Let $h_{n-1}$ be an arbitrary element of the group $Y$. Substitute $u+b_{n-1}h_{n-1}$ for $u$ and $v+a_{n-1}h_{n-1}$ for $v$ in equation (8). Subtracting equation (8) from the resulting equation we obtain

[TABLE]

where $l_{n-1,j}=a_{j}b_{n-1}+b_{j}a_{n-1}$, $m_{n-1,j}=a_{j}b_{n-1}-b_{j}a_{n-1}$.

The right-hand side of equation (9) does not contain the functions $\psi_{n}$ and $\psi_{n-1}$. Arguing as above we get through $n$ steps the equation

[TABLE]

where $l_{ij}=a_{j}b_{i}+b_{j}a_{i}$.

Set $m_{ij}=a_{j}b_{i}-b_{j}a_{i}$ for all $i,j$.

Let $k_{n}$ be an arbitrary element of the group $Y$. Substitute $u+b_{n}k_{n}$ for $u$ and $v-a_{n}k_{n}$ for $v$ in equation (10). Subtracting equation (10) from the resulting equation we obtain

[TABLE]

Note that the left-hand side of equation (11) does not contain the function $\psi_{n}$. Arguing as above we get through $n-1$ steps the equation

[TABLE]

Put $v=0$ in (12). We have

[TABLE]

We divide the proof into some steps. First, we prove the theorem, assuming that all integers $b_{i}a_{j}-b_{j}a_{i}$ are admissible for all $i\neq j$ for the group $X$. Then we prove that the case when not all integers $b_{i}a_{j}-b_{j}a_{i}$ are admissible for all $i\neq j$ for the group $X$ can be reduced to the case when all integers $b_{i}a_{j}-b_{j}a_{i}$ are admissible for all $i\neq j$ for the group $X$.

1. Let all integers $b_{i}a_{j}-b_{j}a_{i}$ be admissible for all $i\neq j$ for the group $X$.

(i) Let $X$ be a torsion-free group, i.e. $X_{(m)}=\{0\}$ for all $m\in\mathbb{Z}$, $m\neq 0$. This implies that $\overline{Y^{(m)}}=Y$ for all integers $m,\ m\neq 0$ ([9, 24.41]). Particularly, $\overline{Y^{(a_{j})}}=Y$, $j=1,2,...,n$. Since integers $l_{ij},m_{ij}$ are admissible, it means for torsion-free groups that $l_{ij}\neq 0,m_{ij}\neq 0$. Therefore, $\overline{Y^{(l_{ij})}}=Y$, $\overline{Y^{(m_{ij})}}=Y$, $j=1,2,...,n$. Taking into account that $h_{j},k_{j}$ are arbitrary elements of the group $Y$, (13) yields that the function $\psi_{1}(y)$ satisfy the equation

[TABLE]

So $\psi_{1}(y)$ is a continuous polynomial on the group $Y$. It follows from Lemma 3 that $\mu_{1}\in\Gamma(X)$. Arguing as above we prove that $\mu_{j}\in\Gamma(X)$ ($j=2,\dots,n$).

(ii) Let $X=X_{(p)}$, where $p$ is a prime number ($p\neq 2$).

Since integers $a_{j},b_{j}$, $j=1,...n$, $b_{i}a_{j}\pm b_{j}a_{i}$ are admissible for the group $X$, we get that endomorphisms $f_{a_{j}},f_{b_{j}},f_{b_{i}a_{j}\pm b_{j}a_{i}}$ are automorphisms. It means that $Y^{(a_{j})}=Y$, $Y^{(b_{j})}=Y$, $Y^{(b_{i}a_{j}\pm b_{j}a_{i})}=Y$. Taking into account that $h_{j},k_{j}$ are arbitrary elements of $Y$, it follows from the form of integers $l_{ij},m_{ij}$ and (13) that the function $\psi_{1}(y)$ satisfy equation (14). Arguing as in case 1(i), we get that all $\mu_{j}\in\Gamma(X)$. Since that component of zero of the group $X$ is equal to zero, we have in case (ii) that $\Gamma(X)=D(X)$.

2. Suppose now that for some $i,j$ integers $b_{i}a_{j}-b_{j}a_{i}$ are admissible for $X$ and for some $i,j$ integers $b_{i}a_{j}-b_{j}a_{i}$ are not admissible for $X$.

(i) Let $X$ be a torsion-free group. In this case if integers $b_{i}a_{j}-b_{j}a_{i}$ are not admissible for $X$ then $b_{i}a_{j}-b_{j}a_{i}=0$ for some $i,j$.

Renumbering random variables, we can assume that

[TABLE]

where $1\leq r_{1}<r_{2}<\dots<r_{k}<n$, and $c_{i}\neq c_{j}$ for all $i\neq j$.

Consider integers

[TABLE]

and random variables

[TABLE]

Put

[TABLE]

Since $L_{1}^{\prime}=b_{1}b_{r_{1}+1}\dots b_{r_{k}+1}L_{1}$, $L_{2}^{\prime}=L_{2}$, it is obvious that the conditional distribution of $L_{2}^{\prime}$ given $L_{1}^{\prime}$ is also symmetric. It follows from (15) and (16) that the coefficients of the linear forms $L_{1}^{\prime}$ and $L_{2}^{\prime}$ satisfy conditions $a_{1}d_{0}\neq 0,a_{r_{1}+1}d_{1}\neq 0,\cdots,a_{r_{k}+1}d_{k}\neq 0$, $a_{1}d_{0}\pm a_{r_{j}+1}d_{j}\neq 0,a_{r_{i}+1}d_{i}\pm a_{r_{j}+1}d_{j}\neq 0$. It follows from case 1(i) that all $\zeta_{j}$ have Gaussian distributions. Applying Lemma 2 we get that $b_{1}\xi_{1},\cdots,b_{n}\xi_{n}$ have Gaussian distributions. Since $X$ is a torsion-free group, the endomorphisms $f_{b_{1}},\cdots,f_{b_{n}}$ are monomorphisms. Now it is easy to verify that the random variables $\xi_{1},\cdots,\xi_{n}$ have Gaussian distributions.

(ii) Let $X=X_{(p)}$, where $p$ is a prime number ($p>2$). Since integers $a_{j},b_{j}$ are admissible for $X$, the endomorphisms $f_{a_{1}},\cdots,f_{a_{n}},f_{b_{1}},\cdots,f_{b_{n}}$ are automorphisms. Therefore we can take into consideration new random variables and assume that all $a_{j}=1$ and all $b_{j}\in\{1,2,\dots,p-1\}$. Then we have $b_{i}-b_{j}=0$ for some $i,j$. Arguing as in case 2(i), we get that all $\xi_{1},\cdots,\xi_{n}$ are degenerated.

3. Let all integers $b_{i}a_{j}-b_{j}a_{i}$ be not admissible for $X$. Then $b_{1}L_{1}=a_{1}L_{2}$. It is obvious that all random variables $\xi_{1},\cdots,\xi_{n}$ are degenerated.

(II) Suppose now that $X$ is not isomorphic to any of the groups mentioned in (I). Then $X$ contains an element $x_{0}$ of order $p$, where $p$ is a prime number and $X^{(p)}\neq\{0\}$. Let $M$ be a subgroup of $X$ generated by the element $x_{0}$. Then $M\cong\mathbb{Z}(p)$. Let $\xi_{1}$ and $\xi_{2}$ be independent identically distributed random variables with values in $M$ and with a distribution

[TABLE]

where $0<a<1$. We consider the distribution $\mu$ as a distribution on the group $X$. Taking into account (2), we have that

[TABLE]

We consider the linear forms $L_{1}=p\xi_{1}-\xi_{2}$ and $L_{2}=\xi_{1}+p\xi_{2}$. Since $X^{(p)}\neq\{0\}$ and $X_{(p)}\neq\{0\}$, the integers $p,p^{2}-1$ are admissible for $X$. By Lemma 1 the conditional distribution of $L_{2}$ given $L_{1}$ is symmetric if and only if the characteristic function $\widehat{\mu}(y)$ satisfy equation (4) which takes the form

[TABLE]

Since $\sigma(\mu)\subset M$, we have $\widehat{\mu}(y)=1$ for $y\in A(Y,M)$ and $\widehat{\mu}(y)$ is $A(Y,M)$-invariant, i.e. $\widehat{\mu}(y)$ takes a constant value on each coset of the group $Y$ with respect to $A(Y,M)$. Then equation (19) induces an equation on the factor-group

[TABLE]

Since $Y/A(Y,M)\approx\mathbb{Z}(p)$, equation (20) is transformed into the equality. Hence by Lemma 1 the conditional distribution of $L_{2}$ given $L_{1}$ is symmetric. $\blacksquare$

Now we describe locally compact Abelian groups for which the group analogue of the Heyde theorem takes place without assumption that the characteristic functions of the considering random variables do not vanish. Note that the obtained class of groups is changed (compare with Theorem 1).

Theorem 2. Let $X$ be a second countable locally compact Abelian group such that $X\neq X_{(2)}$. Then the following statements hold:

(I) Let $\xi_{j},\ j=1,2,...,n,\ n\geq 2,$ be independent random variables with values in $X$ and distributions $\mu_{j}$. Consider the linear forms $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ and $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$, where the coefficients $a_{j},\ b_{j}$ are admissible integers for $X$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$. Assume that the conditional distribution of $L_{2}$ given $L_{1}$ is symmetric. Then the following statements hold:

$(i)$* If $X=\mathbb{R}^{n}\times D$, where $n\geq 0$ and $D$ is a torsion-free group, then all $\mu_{j}\in\Gamma(X)$;*

$(ii)$* If $X=X_{(3)}$ then all $\mu_{j}\in D(X)$.*

(II) If $X$ is not isomorphic to any of the groups mentioned in (I), then there exist independent random variables $\xi_{j},j=1,2,...,n,n\geq 2,$ with values in $X$ and distributions $\mu_{j}$, and admissible integers $a_{j},\ b_{j}$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$, such that the conditional distribution of $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$ given $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ is symmetric, but all $\mu_{j}\not\in\Gamma(X)*I(X)$.

Remark 2. In Theorem 2 we described all groups on which the symmetry of the conditional distribution of one linear form given another implies that all distributions belong to the class $\Gamma(X)*I(X)$. In fact if a distribution belongs to the class $\Gamma(X)*I(X)$ then it belongs to the class $\Gamma(X)$.

To prove Theorem 2 we need some lemmas.

Lemma 4. Let $K$ be a connected compact Abelian group and $\mu_{j}$ are distributions on the group $K^{*}$. Assume that the characteristic functions $\widehat{\mu}_{j}(y)$ satisfy equation $(\ref{1.1})$ on the group $K$ and $\widehat{\mu}_{j}(y)\geq 0$. If $a_{j}b_{j}\neq 0$, then all $\widehat{\mu}_{j}(y)=1$, $y\in K$.

The proof of Lemma 4 is carried out in the same way as the proof of Lemma 1 of the article [3], but it is based on the Heyde theorem. For completeness of the statement we will give this proof in the article.

Proof of Lemma 4. Two cases are possible.

1. $K\not\approx\mathbb{T}$. Since $K$ is a connected compact group, there exists a continuous monomorphism $f:\mathbb{R}\rightarrow K$ such that $\overline{f(\mathbb{R})}=K$ ([9, §25.18]). Consider the restriction of equation (4) to $f(\mathbb{R})$. It follows from the Heyde theorem that $\widehat{\mu}_{j}(f(t))=\exp\{-\lambda_{j}t^{2}\}$, where $\lambda_{j}\geq 0$ for $t\in\mathbb{R}$.

Let $U$ be a neighborhood of zero of $K$. Since $f$ is a monomorphism and $\overline{f(\mathbb{R})}=K$, there exists a sequence $t_{n}\longrightarrow+\infty$ such that $f(t_{n})\in U$ for all $n$. If $\lambda_{j}>0$ for some $j$ then $\widehat{\mu}_{j}(f(t))=\exp\{-\lambda_{j}t^{2}\}\longrightarrow 0$ for $t_{n}\longrightarrow+\infty$, which contradicts to the continuity of the function $\widehat{\mu}_{j}(y)$ at zero. We have that all $\lambda_{j}=0$. So, $\widehat{\mu}_{j}(f(t))=1$ for $t\in\mathbb{R}$. Since $\overline{f(\mathbb{R})}=K$, we have $\widehat{\mu}_{j}(y)=1$ for $y\in K$.

2. $K\approx\mathbb{T}$. In this case the characteristic functions $\widehat{\mu}_{j}(y)$ are $2\pi$-periodic and satisfy equation (4). It follows from the Heyde theorem that $\widehat{\mu}_{j}(y)$ are the characteristic functions of Gaussian distributions, i.e. $\widehat{\mu}_{j}(y)=\exp\{-\lambda_{j}y^{2}\}$, where $\lambda_{j}\geq 0$. It follows from $2\pi$-periodicity that $\lambda_{j}=0$. So, $\widehat{\mu}_{j}(y)=1$ for $y\in K$. $\blacksquare$

Lemma 5. Let $X=\Delta_{p}$. Then there exist independent random variables $\xi_{1},\xi_{2}$ with values in $X$ and distributions $\mu_{1},\mu_{2}$ such that the conditional distribution of $L_{2}=\xi_{1}+p\xi_{2}$ given $L_{1}=p\xi_{1}-\xi_{2}$ is symmetric and $\mu_{1},\mu_{2}\not\in I(X)$.

Proof. Since $X=\Delta_{p}$, we have $Y\approx\mathbb{Z}(p^{\infty})$. We can assume without loss of generality that $Y=\mathbb{Z}(p^{\infty})$. Consider the distribution (17) on $\mathbb{Z}(p)$. Note that $Y_{(p)}\approx\mathbb{Z}(p)$. Obviously,

[TABLE]

is a characteristic function on the group $Y_{(p)}$. Consider the function

[TABLE]

where $0<a<1$. It follows from [10, §32.43] that the function $f(y)$ is positive definite. By the Bochner theorem ([10, §33.3]) the function $f(y)$ is the characteristic function of a distribution $\nu$ on $X$. Put $\mu_{1}=\mu_{2}=\nu$ and verify that the conditional distribution of $L_{2}=\xi_{1}+p\xi_{2}$ given $L_{1}=p\xi_{1}-\xi_{2}$ is symmetric. By Lemma 1 it suffices to show that the characteristic function $f(y)$ satisfies equation (19). Obviously, if $u,v\in Y_{(p)}$ then equation (19) is satisfied. If either $u\in Y_{(p)}$, $v\not\in Y_{(p)}$, or $u\not\in Y_{(p)}$, $v\in Y_{(p)}$, both sides of equation (19) are equal to zero. Let $u,v\not\in Y_{(p)}$. If the left-hand side of equation (19) is not equal to zero, then $pu+v,-u+pv\in Y_{(p)}$. We conclude from this that $(p^{2}+1)v\in Y_{(p)}$. Hence $v\in Y_{(p)}$ contrary to the assumption. Thus, the left-hand side of equation (19) is equal to zero. Arguing in the same way, we get that the right-hand side of equation (19) is equal to zero. We get that the function $f(y)$ satisfies equation (19). It is clear that $\nu$ can be chosen such that $\mu\not\in I(X)$. $\blacksquare$

Remark 3. If in Lemma 5 independent random variables have distributions with non-vanishing characteristic functions then Theorem 1 implies that the distributions are degenerated.

Lemma 6. Let $X=\mathbb{Z}(p)$, where $p$ is a prime number ($p>3$). There exist independent random variables $\xi_{1},\xi_{2},\xi_{3},\xi_{4}$ with values in $X$ and distributions $\mu_{1},\mu_{2},\mu_{3},\mu_{4}$ such that the conditional distribution of $L_{2}=\xi_{1}+\xi_{2}+2\xi_{3}+2\xi_{4}$ given $L_{1}=\xi_{1}+\xi_{2}+\xi_{3}+\xi_{4}$ is symmetric and $\mu_{1},\mu_{2},\mu_{3},\mu_{4}\not\in I(X)$.

Proof. Since $X=\mathbb{Z}(p)$, we have $Y\approx\mathbb{Z}(p)$. Choose nonzero elements $y_{1},y_{2}\in Y$such that $y_{1}\neq\pm y_{2}$. Let $\nu_{1},\nu_{2}$ be distributions in $X$ with the densities $\rho_{1}(x)=1+{1\over 2}{\rm Re}(x,y_{1})$, $\rho_{2}(x)=1+{1\over 2}{\rm Re}(x,y_{2})$ with respect to the Haar distribution $m_{X}$ respectively. Then

[TABLE]

Put $\mu_{1}=\mu_{3}=\nu_{1}$, $\mu_{2}=\mu_{4}=\nu_{2}$ and verify that the conditional distribution of $L_{2}=\xi_{1}+\xi_{2}+2\xi_{3}+2\xi_{4}$ given $L_{1}=\xi_{1}+\xi_{2}+\xi_{3}+\xi_{4}$ is symmetric. By Lemma 1 it suffices to show that the characteristic functions of distributions $\mu_{1},\mu_{2},\mu_{3},\mu_{4}$ satisfy equation (4), which takes the form

[TABLE]

Since $y_{1}\neq\pm y_{2}$, taking into account (2) we have $\widehat{\nu}_{1}(y)\widehat{\nu}_{2}(y)=\widehat{m}_{X}(y)$. Equation (22) is equivalent to the equation

[TABLE]

Obviously, equation (23) is satisfied if either $u=0$ or $v=0$. Let $u\neq 0$, $v\neq 0$. If the left-hand side of equation (23) is not equal to zero, then $u+v=0$, $u+2v=0$. We conclude from this that $v=0$ contrary to the assumption. Thus, the left-hand side of equation (23) is equal to zero. Arguing in the same way, we get that the right-hand side of equation (23) is equal to zero. We get that the function $\widehat{m}_{X}(y)$ satisfies equation (23). It is clear that $\nu_{1},\nu_{2}\not\in I(X)$. $\blacksquare$

Proof of Theorem 2. (I)(i) If $X=\mathbb{R}^{n}\times D$, then $Y\approx\mathbb{R}^{n}\times K$, where $K$ is a connected compact group. We can assume without loss of generality that $Y=\mathbb{R}^{n}\times K$.

Lemma 1 implies that the characteristic functions of distributions $\mu_{j}$ satisfy equation (4). Reasoning as in beginning of the proof of Theorem 1, we can assume that $\widehat{\mu}_{j}(y)\geq 0$.

Consider the restriction of equation (4) to $K$. It follows from Lemma 4 that $\widehat{\mu}_{j}(y)=1$ on $K$. Then $\sigma(\mu_{j})\subset A(X,K)=\mathbb{R}^{n}$. We consider the restriction of equation (4) to each one-dimensional subspace and obtain from the Heyde theorem that all restriction of the characteristic functions $\widehat{\mu}_{j}(y)$ are the characteristic functions of Gaussian distributions. It follows from this that $\mu_{j}\in\Gamma(X)$.

(ii) If $X=X_{(3)}$ then we can assume without loss of generality that $a_{j},b_{j}$ are equal to $\pm 1$. Since all integers $b_{i}a_{j}+b_{j}a_{i}$ are admissible, we have either $L_{1}=L_{2}$ or $L_{1}=-L_{2}$. It easily follows from this that all random variables $\xi_{1},\cdots,\xi_{n}$ are degenerated.

(II) 1. By the structure theorem for locally compact Abelian groups $X\approx\mathbb{R}^{n}\times G$, where $G$ contains an open compact subgroup $H$ ([9, 24.30]). Suppose that $X$ is a torsion-free group. Since $X$ is not as in case (I), we have $H\neq\{0\}$. A compact torsion-free group is topologically isomorphic to the group

[TABLE]

where $\Sigma_{a}^{*}=\mathbb{Q}$, $\mathfrak{n},\mathfrak{n}_{p}$ are cardinal numbers (see [9, 25.8]). Note that for each prime number $p$ a group $\Sigma_{a}$ contains a subgroup $\Delta_{p}$. It follows from the fact that the factor-group $\mathbb{Q}/\mathbb{Z}$ contains a subgroup $\mathbb{Z}(p^{\infty})$ for each prime number $p$. Then the statement of Theorem 2 follows from Lemma 5.

2. Suppose that $X$ contains an element $x_{0}$ of order $p$, where $p$ is a prime number and $X\neq X_{(p)}$. Then the statement of Theorem 2 follows part (II) of Theorem 1.

3. Suppose that $X=X_{(p)}$, where $p>3$. Then the statement of Theorem 2 follows from Lemma 6.

Theorem 2 is proved. $\blacksquare$

4 The Heyde theorem for locally compact Abelian groups for Q-independent

random variables

In the article [13] A.M.Kagan and G.J.Székely introduced a notion of $Q$-independence of random variables which generalizes the notion of independence of random variables. Then in [7] G.M. Feldman in a natural way introduced a notion of $Q$-independence of random variables taking values in a locally compact Abelian group. He proved that if we consider $Q$-independence instead of independence, then the group analogue of the Cramér theorem about decomposition of a Gaussian distribution and some group analogues of the the Skitovich–Darmois and Heyde theorems hold true for the same classes of groups. These studies were continued in [17]. Some analogues of characterization theorems for Q-independent random variables with values in Banach spaces were studied in [18]. We prove in this section that the results of §3 hold true if instead of independence $Q$-independence is considered.

Let $\xi_{1},\dots,\xi_{n}$ be random variables with values in the group $X$. We say that the random variables $\xi_{1},\dots,\xi_{n}$ are $Q$-independent if the characteristic function of the vector $(\xi_{1},\dots,\xi_{n})$ can be represented in the form

[TABLE]

[TABLE]

where $q(y_{1},\dots,y_{n})$ is a continuous polynomial on the group $Y^{n}$ such that $q(0,\dots,0)=0$.

As is well known, any continuous polynomial is equal to the constant on compact elements (see e.g. [6, §5.7]). If the group $Y$ consists only of compact elements then the connected component of zero of the group $X$ is equal to zero (see e.g. [9, §24.17]). Thus, independence and Q-independence of random variables are equivalent on groups $X$ whose connected component of zero is equal to zero.

We prove that Theorem 1 remains true if we change the condition of independence of $\xi_{1},\dots,\xi_{n}$ for $Q$-independence. The following statement holds true.

Theorem 3. Let $X$ be a second countable locally compact Abelian group such that $X\neq X_{(2)}$. Then the following statements hold:

(I) Let $\xi_{j},\ j=1,2,...,n,\ n\geq 2,$ be Q-independent random variables with values in $X$ and with distributions $\mu_{j}$ with non-vanishing characteristic functions. Consider the linear forms $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ and $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$, where the coefficients $a_{j},\ b_{j}$ are admissible integers for $X$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$. Assume that the conditional distribution of $L_{2}$ given $L_{1}$ is symmetric. Then the following statements hold:

$(i)$* If $X$ is a torsion-free group then all $\mu_{j}\in\Gamma(X)$;*

$(ii)$* If $X=X_{(p)}$, where $p$ is a prime number ($p>2$), then all $\mu_{j}\in D(X)$.*

(II) If $X$ is not isomorphic to any of the groups mentioned in (I), then there exist Q-independent random variables $\xi_{j},j=1,2,...,n,n\geq 2,$ with values in $X$ and distributions $\mu_{j}$ with non-vanishing characteristic functions, and admissible integers $a_{j},\ b_{j}$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$ such that the conditional distribution of $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$ given $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ is symmetric, but all $\mu_{j}\not\in\Gamma(X)$.

To prove Theorem 3 we need the following lemma.

Lemma 7. Let $\xi_{j},\ j=1,2,...,n,\ n\geq 2,$ be Q-independent random variables with values in a second countable locally compact Abelian group $X$ and with distributions $\mu_{j}$. Let $a_{j},\ b_{j}$ be integers. The conditional distribution of the linear form $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$ given $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ is symmetric if and only if the characteristic functions $\widehat{\mu}_{j}(y)$ satisfy the equation

[TABLE]

where $q(u,v)$ is a continuous polynomial on the group $Y^{2}$, $q(0,0)=0$.

Proof. The condition of the symmetry of the conditional distribution of the linear form $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$ given $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ is equivalent to equation (5).

Taking into account the Q-independence of the random variables $\xi_{j}$, we have

[TABLE]

[TABLE]

where $q_{0}(y_{1},\dots,y_{n})$ is a continuous polynomial on the group $Y^{n}$, $q_{0}(0,\dots,0)=0$. Put $q(u,v)=q_{0}(a_{1}u-b_{1}v,\dots,a_{n}u-b_{n}v)-q_{0}(a_{1}u+b_{1}v,\dots,a_{n}u+b_{n}v)$. Taking into account that $\widehat{\mu}_{j}(y)=\mathbf{E}[(\xi_{j},y)]$, we obtain equation (25). $\blacksquare$

Proof of Theorem 3. (I) Lemma 7 implies that the characteristic functions of distributions $\mu_{j}$ satisfy equation (25). Reasoning as in the beginning of the proof of Theorem 1, we can assume that $\widehat{\mu}_{j}(y)>0$ and the polynomial $q(u,v)$ is real-valued.

We will show that $\widehat{\mu}_{j}(y)$ are the characteristic functions of Gaussian distributions. Put $\psi_{j}(y)=-\log\widehat{\mu}_{j}(y)$. It follows from (25) that

[TABLE]

We use the finite-difference method in the same manner as in Theorem 1 to solve equation (27). We will retain the notation from the proof of Theorem 1. In fact we add one more step. Instead of equation (12), we get through $2n-1$ steps the equation

[TABLE]

[TABLE]

Since $q(u,u)$ is a polynomial, we have

[TABLE]

for some $l$ and arbitrary elements $h$ and $k$ of $Y$.

We substitute $v=0$ in (4) and apply the operator $\Delta^{l+1}_{(h,k)}$ to (4). Taking into account (29), we obtain

[TABLE]

We complete the proof of Theorem 2 in the same way as Theorem 1.

(II) Since independent random variables are Q-independent, this part of Theorem 3 follows from part (II) of Theorem 1. $\blacksquare$

We prove that Theorem 2 remains true if we change the condition of independence of $\xi_{1},\dots,\xi_{n}$ for $Q$-independence. The following statement holds true.

Theorem 4. Let $X$ be a second countable locally compact Abelian group such that $X\neq X_{(2)}$. Then the following statements hold:

(I) Let $\xi_{j},\ j=1,2,...,n,\ n\geq 2,$ be Q-independent random variables with values in $X$ and with distributions $\mu_{j}$. Consider the linear forms $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ and $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$, where the coefficients $a_{j},\ b_{j}$ are admissible integers for $X$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$. Assume that the conditional distribution of $L_{2}$ given $L_{1}$ is symmetric. Then the following statements hold:

$(i)$* If $X=\mathbb{R}^{n}\times D$, where $n\geq 0$ and $D$ is a torsion-free group, then all $\mu_{j}\in\Gamma(X)$;*

$(ii)$* If $X=X_{(3)}$ then all $\mu_{j}\in D(X)$.*

(II) If $X$ is not isomorphic to any of the groups mentioned in (I), then there exist Q-independent random variables $\xi_{j},j=1,2,...,n,n\geq 2,$ with values in $X$ and distributions $\mu_{j}$, and admissible integers $a_{j},\ b_{j}$ such that $b_{i}a_{j}+b_{j}a_{i}$ are admissible integers for $X$ for all $i,j$ such that the conditional distribution of $L_{2}=b_{1}\xi_{1}+\cdots+b_{n}\xi_{n}$ given $L_{1}=a_{1}\xi_{1}+\cdots+a_{n}\xi_{n}$ is symmetric, but all $\mu_{j}\not\in\Gamma(X)*I(X)$.

Proof. (I)(i) If $X=\mathbb{R}^{n}\times D$, then $Y\approx\mathbb{R}^{n}\times K$, where $K$ is a connected compact group. We can assume without loss of generality that $Y=\mathbb{R}^{n}\times K$.

Lemma 7 implies that the characteristic functions of distributions $\mu_{j}$ satisfy equation (25). Reasoning as in the beginning of the proof of Theorem 1, we can assume that $\widehat{\mu}_{j}(y)>0$ and the polynomial $q(u,v)$ is real-valued.

Consider the restriction of equation (25) to $K$. Since each continuous polynomial is equal to a constant on compact elements, the restriction of equation (25) to $K$ coincides with equation (4). It follows from Lemma 4 that $\widehat{\mu}_{j}(y)=1$ on $K$. Then $\sigma(\mu_{j})\subset A(X,K)=\mathbb{R}^{n}$. Thus, we reduced the proof to the consideration of equation (25) on $\mathbb{R}^{n}$. It follows from [18] that all $\mu_{j}\in\Gamma(X)$.

(ii) If $X=X_{(3)}$ then the connected component of zero of $X$ is equal to zero. The independence and the Q-independence are coincides on such groups. Therefore the statement of Theorem 4 in this case follows from Theorem 1.

(II) Since independent random variables are Q-independent, this part of Theorem 4 follows from part (II) of Theorem 2. $\blacksquare$