Generalized Pólya’s theorem on connected locally

compact Abelian groups of dimension 1

G.M. Feldman

Abstract. According to the generalized Pólya theorem, the Gaussian
distribution on the real line is characterized by the property of
equidistribution of a monomial and a linear form of independent identically
distributed random variables. We give a complete description of $a$-adic
solenoids for which an analog of this theorem is true. The proof of the main
theorem is reduced to solving some functional equation in the class of
continuous positive definite functions on the character group of an $a$-adic
solenoid.

Mathematics Subject Classification. 43A25, 43A35, 60B15, 62E10.

Keywords. $a$-adic solenoid, topological automorphism, Gaussian distribution,
Haar distribution.

##  1 Introduction

According to the classical characterization theorems of mathematical
statistics, the Gaussian distribution on the real line is characterized by the
independence of two linear forms of independent random variables (the
Skitovich–Darmois theorem), as well as by the symmetry of the conditional
distribution of one linear form given another (the Heyde theorem). A number of
papers have been devoted to generalizing of these theorems to various
algebraic structures, in particular, to locally compact Abelian groups (see
e.g. [5, 6, 8, 13, 12, 14, 9, 10, 11, 18], and also [7, Chapters IV–VI]). In
so doing, the coefficients of linear forms are topological automorphisms of
the group. In this note, we consider a group analog of the generalized Pólya
theorem, in which the Gaussian distribution on the real line is characterized
by the property of equidistribution of a monomial and a linear form of
independent identically distributed random variables. We study the generalized
Pólya theorem on $a$-adic solenoids. This is an important class of locally
compact Abelian groups, since each connected locally compact Abelian group of
dimension 1 is topologically isomorphic to either the additive group of real
numbers, or the circle group, or an $a$-adic solenoid. Furthermore, the
coefficients of a monomial and a linear form, as in the group analogs of the
Skitovich–Darmois and Heyde theorems, are topological automorphisms of the
$a$-adic solenoid. As far as we know, such problem for groups has not been
considered earlier. Note also that in the list of unsolved problems in [17,
Ch. 14] the problem of constructing a theory of equidistributed linear forms
on algebraic structures was formulated. A step in this direction is this note.

Recall the generalized Pólya theorem.

Theorem A ([17, §13.7]). Let $\xi_{j}$, $j=1,2,\dots,n$, $n\geq 2$, be
independent identically distributed random variables. Let $\alpha_{j}$ be
nonzero real numbers such that

$\alpha_{1}^{2}+\dots+\alpha_{n}^{2}=1.$ (1)

If $\xi_{j}$ and $\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are identically
distributed, then $\xi_{j}$ are Gaussian.

If we denote by $f(s)$ the characteristic function of $\xi_{j}$ it is easy to
see that the generalized Pólya theorem is equivalent to the statement that all
solutions of the equation

$f(s)=f(\alpha_{1}s)\cdots f(\alpha_{n}s),\ \ s\in\mathbb{R},$

in the class of normed continuous positive definite functions are of the form
$f(s)=\exp\\{-\sigma s^{2}+i\beta s\\}$, where $\sigma\geq 0$,
$\beta\in\mathbb{R}$.

The first theorem of this kind, namely when $n=2$ and
$\alpha_{1}=\alpha_{2}={1\over\sqrt{2}}$, was proved by G. Pólya in 1923.
Subsequently, the characterization of the Gaussian distribution on the real
line by the property of equidistribution of two linear forms of independent
identically distributed random variables were studied by I. Marcinkiewicz, Yu.
V. Linnik, A.A. Zinger and others (see [17, Chapter 2]).

Let $X$ be a second countable locally compact Abelian group. We will consider
only such groups, without mentioning it specifically. Let ${\rm Aut}(X)$ be
the group of topological automorphisms of the group $X$, and $I$ be 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 $y\in Y$ at an element
$x\in X$. For a closed subgroup $K$ of $X$, denote by $A(Y,K)=\\{y\in
Y:(x,y)=1$ for all $x\in K\\}$ its annihilator. If $\alpha\in{\rm Aut}(X)$,
then define the adjoint automorphism $\tilde{\alpha}\in{\rm Aut}(Y)$ by the
formula $(x,\tilde{\alpha}y)=(\alpha x,y)$ for all $x\in X$, $y\in Y$. Denote
by $\mathbb{R}$ the group of real numbers, by $\mathbb{Z}$ the group of
integers and by $\mathbb{T}$ the circle group. Let $n$ be an integer, $n\neq
0$. Denote by $f_{n}:X\mapsto X$ an endomorphism of the group $X$ defined by
the formula $f_{n}x=nx$, $x\in X$.

Let ${\rm M}^{1}(X)$ be the convolution semigroup of probability distributions
on the group $X$. For $\mu\in{\rm M}^{1}(X)$ denote by

$\hat{\mu}(y)=\int_{X}(x,y)d\mu(x),\ \ y\in Y,$

the characteristic function (Fourier transform) of the distribution $\mu$, and
define the distribution $\bar{\mu}\in{\rm M}^{1}(X)$ by the rule
$\bar{\mu}(B)=\mu(-B)$ for all Borel subsets $B$ in $X$. We have
$\hat{\bar{\mu}}(y)=\overline{\hat{\mu}(y)}$.

A distribution $\gamma\in{\rm M}^{1}(X)$ is called Gaussian (see [19, Chapter
IV, §6]) if its characteristic function can be represented in the form

$\hat{\gamma}(y)=(x,y)\exp\\{-\varphi(y)\\},\ \ y\in Y,$

where $x\in X$, and $\varphi(y)$ is a continuous non-negative function on the
group $Y$ satisfying the equation

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

Note that, in particular, the degenerate distributions are Gaussian. Denote by
$\Gamma(X)$ the set of Gaussian distributions on $X$.

Let $K$ be a compact subgroup of the group $X$. Denote by $m_{K}$ the Haar
distribution on $K$. The characteristic function of the distribution $m_{K}$
is of the form

$\hat{m}_{K}(y)=\begin{cases}1,&\text{\ if\ }\ y\in A(Y,K),\\\ 0,&\text{\ if\
}\ y\not\in A(Y,K).\end{cases}$ (2)

##  2 Main theorem

Let $a$=$(a_{0},a_{1},a_{2},\dots)$, where all $a_{j}\in{\mathbb{Z}}$,
$a_{j}>1$, and let $\Delta_{{\text{\boldmath$a$}}}$ be the group of $a$-adic
integers ([15, (10.2)]). As a set $\Delta_{{\text{\boldmath$a$}}}$ coincides
with the Cartesian product $\mathop{\mbox{\rm\bf
P}}\limits_{n=0}^{\infty}\\{0,1,\dots,a_{n}-1\\}$. Put
$\mathbf{u}=(1,0,\dots,0,\dots)\in\Delta_{{\text{\boldmath$a$}}}$. Consider
the group $\mathbb{R}\times\Delta_{{\text{\boldmath$a$}}}$ and denote by $B$ a
subgroup of the group $\mathbb{R}\times\Delta_{{\text{\boldmath$a$}}}$ of the
form $B=\\{(n,n\mathbf{u})\\}_{n=-\infty}^{\infty}$. The factor-group
$\Sigma_{{\text{\boldmath$a$}}}=(\mathbb{R}\times\Delta_{{\text{\boldmath$a$}}})/B$
is called an $a$-adic solenoid. The group $\Sigma_{{\text{\boldmath$a$}}}$ is
compact, connected and has dimension 1 ([15, (10.12), (10.13), (24.28)]). The
character group of the group $\Sigma_{{\text{\boldmath$a$}}}$ is topologically
isomorphic to a discrete additive group $H_{\text{\boldmath$a$}}$ of the
rational numbers of the form

$H_{\text{\boldmath$a$}}=\left\\{{m\over a_{0}a_{1}\cdots a_{n}}:\
n=0,1,\dots;\ m\in{\mathbb{Z}}\right\\}$ (3)

([15, (25.3)]). It is convenient for us to assume that
$H_{\text{\boldmath$a$}}$ is the character group of the group
$\Sigma_{\text{\boldmath$a$}}$. Each topological automorphism $\alpha\in{\rm
Aut}(\Sigma_{\text{\boldmath$a$}})$ is of the following form
$\alpha=f_{p}f_{q}^{-1}$ for some mutually prime $p$ and $q$, where
$f_{p},f_{q}\in{\rm Aut}(\Sigma_{\text{\boldmath$a$}})$. We will identify
$\alpha=f_{p}f_{q}^{-1}$ with the rational number ${p\over q}$. Note that
$\tilde{f}_{n}=f_{n}$. Hence, if $\alpha=f_{p}f_{q}^{-1}\in{\rm
Aut}(\Sigma_{\text{\boldmath$a$}})$, then
$\tilde{\alpha}=f_{p}f_{q}^{-1}\in{\rm Aut}(H_{\text{\boldmath$a$}})$.

The main result of this note is the following theorem.

######  Theorem 2.1.

Let $X=\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid satisfying the
condition:

$(i)$ There is a unique prime number $p$ such that the group $X$ contains no
elements of order $p$.

Let $\alpha_{j}$, $j=1,2,\dots,n$, $n\geq 2$, be topological automorphisms of
the group $X$ such that

$\alpha_{1}^{2}+\dots+\alpha_{n}^{2}=I.$ (4)

Let $\xi_{j}$ be independent identically distributed random variables with
values in $X$ and distribution $\mu$. If $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are identically distributed, then
$\mu=\gamma*m_{K}$, where $\gamma\in\Gamma(X)$, and $K$ is a compact subgroup
of the group $X$. Moreover, $f_{p}(K)=K$.

We note that if $\mu=\gamma*m_{K}$, where $\gamma\in\Gamma(X)$ and $K$ is a
compact subgroup of the group $X$, then $\mu$ is invariant with respect to
$K$, and $\mu$ induces on the factor-group $X/K$ under the natural mapping
$X\rightarrow X/K$ a Gaussian distribution. The condition (4) for $a$-adic
solenoids is an analog of the condition (1) for the real line. Thus, Theorem
2.1 can be considered as an analog for $a$-adic solenoids satisfying the
condition $(i)$ of the generalized Pólya theorem. Observe that group analogs
of the generalized Pólya theorem for locally compact Abelian groups in the
case of equidistribution of a monomial and a linear form with integer
coefficients were studied in [3, §11].

######  Corollary 2.2.

Let $X=\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid satisfying the
condition $(i)$ of Theorem 2.1. Let $\alpha_{j}$, $j=1,2,\dots,n$, $n\geq 2$,
be topological automorphisms of the group $X$ satisfying the condition
$(\ref{1})$. Let $\xi_{j}$ be independent identically distributed random
variables with values in $X$ and distribution $\mu$ with a non-vanishing
characteristic function. If $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are identically distributed, then
$\mu\in\Gamma(X)$.

######  Remark 2.3.

Let $a$=$(a_{0},a_{1},a_{2},\dots)$, and let $\Sigma_{{\text{\boldmath$a$}}}$
be the corresponding $a$-adic solenoid.

It is easy to verify that the following statements are equivalent:

$(i)$ There is a unique prime number $p$ such that
$\Sigma_{{\text{\boldmath$a$}}}$ contains no elements of order $p$;

$(ii)$ There is a unique prime number $p$ such that infinite number of the
numbers $a_{j}$ are divided by $p$.

An example of an $a$-adic solenoid $\Sigma_{{\text{\boldmath$a$}}}$ that
satisfies the condition $(i)$ of Theorem 2.1 is an $a$-adic solenoid
$\Sigma_{{\text{\boldmath$a$}}}$, where $a$=$(p,p,p,\dots)$. Its character
group is topologically isomorphic to a discrete additive group $F$ of the
rational numbers of the form

$F=\left\\{{m\over p^{n}}:\ n=0,1,\dots;\ m\in{\mathbb{Z}}\right\\}.$ (5)

It is easy to see that the following statements are equivalent:

$(\alpha)$ There are topological automorphisms $\alpha_{j}$, $j=1,2,\dots,n$,
$n\geq 2$, of $\Sigma_{{\text{\boldmath$a$}}}$ satisfying the condition
$(\ref{1})$;

$(\beta)$ There is a prime number $p$ such that
$\Sigma_{{\text{\boldmath$a$}}}$ contains no elements of order $p$.

$(\gamma)$ There is a prime number $p$ such that infinite number of the
numbers $a_{j}$ are divided by $p$.

To prove it, note that there is an alternative: either $f_{p}\in{\rm
Aut}(\Sigma_{{\text{\boldmath$a$}}})$ for a prime number $p$ or ${\rm
Aut}(\Sigma_{{\text{\boldmath$a$}}})=\\{\pm I\\}$.

To prove Theorem 2.1 we need two lemmas.

######  Lemma 2.4.

Let $X$ be a locally compact Abelian group, and let $\alpha_{j}$,
$j=1,2,\dots,n$, $n\geq 2$, be topological automorphisms of the group $X$. Let
$\xi_{j}$ be independent identically distributed random variables with values
in $X$ and distribution $\mu$. Then $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are identically distributed if and
only if the characteristic function $\hat{\mu}(y)$ satisfies the equation

$\hat{\mu}(y)=\hat{\mu}(\tilde{\alpha}_{1}y)\cdots\hat{\mu}(\tilde{\alpha}_{n}y),\
\ y\in Y.$ (6)

The proof of the lemma is the same as in the classical case and follows
directly from the fact that $\hat{\mu}(y)={\bf E}[(\xi_{j},y)]$ for all $y\in
Y$, and the independence of the random variables $\xi_{j}$.

The following lemma can be considered as an analog for locally compact Abelian
groups of the well-known Cramér theorem on decomposition of the Gaussian
distribution on the real line ([1], see also [7, Theorem 4.6]).

######  Lemma 2.5.

Let $X$ be a locally compact Abelian group containing no subgroup
topologically isomorphic to the circle group $\mathbb{T}.$ Let
$\gamma\in\Gamma(X)$. If $\gamma=\lambda_{1}*\lambda_{2}$, where
$\lambda_{j}\in{\rm M}^{1}(X)$, then $\lambda_{j}\in\Gamma(X)$, $j=1,2$.

Proof of Theorem 2.1. In order not to complicate the notation we will identify
$Y$ with $H_{\text{\boldmath$a$}}$ and consider $Y$ as a subset of the real
line $\mathbb{R}$. It follows from the definition of the Gaussian distribution
that $\gamma\in\Gamma(X)$ if and only if its characteristic function can be
represented in the form

$\hat{\gamma}(y)=(x,y)\exp\\{-\sigma y^{2}\\},\ \ y\in Y,$

where $x\in X$, $\sigma\geq 0$.

By Lemma 2.4, the characteristic function $\hat{\mu}(y)$ satisfies equation
(6). First, we will prove the theorem assuming that $\hat{\mu}(y)\geq 0$ for
all $y\in Y$. This implies that $\hat{\mu}(-y)=\hat{\mu}(y)$ for all $y\in Y$.

Since $X$ is a connected group, $f_{n}$ is an epimorphism for each nonzero
integer $n$ ([15, (24.25)]). Since $X$ contains no elements of order $p$, we
have that $f_{p}$ is a monomorphism. In view of the fact that $f_{p}$ is an
epimorphism, $f_{p}\in{\rm Aut}(X)$, and for this reason $f_{p}\in{\rm
Aut}(Y)$. It means that both $X$ and $Y$ are groups with unique division by
$p$. Note that in view of the condition $(i)$ of the theorem, if $q$ is a
prime number and $q\neq p$, then $f_{q}\notin{\rm Aut}(X)$. This implies that
each topological automorphism of the group $X$ is of the form $\pm f^{\pm
1}_{p^{m}}$, where $m$ is a non-negative integer. Taking into account (4), we
conclude that each topological automorphism $\alpha_{j}$ in the theorem is
$\alpha_{j}=\pm f^{-1}_{p^{m_{j}}}$, where $m_{j}$ is a natural number, i.e.
$\alpha_{j}$ is a multiplication by a number $\pm p^{-m_{j}}$. Put
$f(y)=\hat{\mu}(y)$. If necessary, changing the numbering of the random
variables $\xi_{j}$ and taking into account that $f(-y)=f(y)$, we can rewrite
equation (6) in the form

$f(y)=(f(p^{-1}y))^{k_{1}}\cdots(f(p^{-l}y))^{k_{l}},\ \ y\in Y,$ (7)

where $k_{j}\geq 0$, $j=1,2,\dots,l$, $k_{1}+\dots+k_{l}=n$. It follows from
(4) that

$\sum_{j=1}^{l}{k_{j}\over p^{2j}}=1.$ (8)

Assume that for each $j$ the inequality $k_{j}\leq p^{j}$ is valid. Then we
have

$\sum_{j=1}^{l}{k_{j}\over p^{2j}}\leq\sum_{j=1}^{l}{1\over p^{j}}<1,$

contrary to (8). Hence, there is $j_{0}$ such that

$k_{j_{0}}>p^{j_{0}}.$ (9)

Taking this into account and the fact that $0\leq f(y)\leq 1$, (7) implies the
inequality

$f(y)\leq(f(p^{-j_{0}}y))^{k_{j_{0}}}\leq(f(p^{-j_{0}}y))^{p^{j_{0}}},\ \ y\in
Y.$ (10)

Set $m=p^{j_{0}}$ and rewrite (10) in the form

$f(my)\leq(f(y))^{m},\ \ y\in Y.$ (11)

Observe that we can assume that $m>2$ in (11). Indeed, if $m=2$, then $p=2$
and $j_{0}=1$. In this case (9) implies that $k_{1}>2$, and in view of (8),
either $k_{1}=4$ or $k_{1}=3$. Let $k_{1}=4$. Then (7) takes the form
$f(y)=(f(2^{-1}y))^{4}$. This implies that $f(4y)=(f(y))^{16}$, and we may
suppose that $m=4$ in (11). Let $k_{1}=3$. Then (7) can be written as
$f(y)=(f(2^{-1}y))^{3}\phi(y)$, where $\phi(y)$ is a characteristic function.
It follows from this that $f(4y)=(f(y))^{9}(\phi(2y))^{3}\phi(4y)$, and we may
also suppose that $m=4$ in (11).

It may be noted that if $\mu=m_{X}$, then the theorem is proved. Therefore,
suppose that $\mu\neq m_{X}$. Then there is an element $y_{0}\in Y$,
$y_{0}\neq 0$, such that $f(y_{0})>0$. Consider a group $F$ of the form (5).
Since $Y$ is a group with unique division by $p$, we can consider on the
subgroup $F$ the function $g(r)=f(ry_{0})$, $r\in F$. Verify that (11) implies
the uniform continuity of the function $g(r)$ in the topology induced on $F$
by the topology of $\mathbb{R}$. Taking into account that $g(r)$ is a positive
definite function on $F$, for this it is enough to show that the function
$g(r)$ is continuous at zero. We will follow the scheme of the proof of Lemma
10 in [2] and Lemma 8 in [4].

Obviously, it follows from (11) that $f(m^{k}y_{0})\leq(f(y_{0}))^{m^{k}}$ for
all natural $k$. Hence,

$f(m^{-k}y_{0})\geq(f(y_{0}))^{1/m^{k}}.$

It implies that

$\textstyle g\left({1\over m^{k}}\right)\geq(g(1))^{1/m^{k}}.$ (12)

Let $a>0$. Note that for all $x>0$ the inequality $1-e^{-ax}\leq ax$ holds.
Taking this into account we get from (12) that

$\textstyle 1-g\left({1\over m^{k}}\right)\leq{C\over m^{k}},$ (13)

where $C=-\ln g(1)$.

It is obvious that for any arbitrary real-valued characteristic function
$f(y)$ the inequality

$1-f(y_{1}+y_{2})\leq 2[(1-f(y_{1}))+(1-f(y_{2}))],\ \ y_{1},y_{2}\in Y,$ (14)

is fulfilled. By induction, this implies the inequality

$1-f(y_{1}+\dots+y_{q})\leq\sum_{j=1}^{q}2^{j}(1-f(y_{j})),\ \ y_{j}\in Y.$
(15)

Let $i$ be an integer such that $0<i<m^{k}$. We have

${i\over m^{k}}={i_{1}\over m}+\dots+{i_{k}\over m^{k}},$ (16)

where $i_{l}$ are integers such that $0\leq i_{l}<m$, $l=1,\dots,k$.
Substituting $q=i_{l}$, $y_{1}=\dots=y_{i_{l}}={y_{0}\over m^{l}}$ in (15), we
get

$\textstyle{1-g\left({i_{l}\over
m^{l}}\right)\leq\sum\limits_{j=1}^{i_{l}}2^{j}\left(1-g\left({1\over
m^{l}}\right)\right)\leq 2^{m}\left(1-g\left({1\over m^{l}}\right)\right)}.$
(17)

Suppose that

${i\over m^{k}}<{1\over m^{s}}.$ (18)

It follows from this that $i_{l}=0$ for all $l\leq s$ in (16). In this case,
taking into account (13), (16) and (17), we find from (15)

$\textstyle{1-g\left({i\over m^{k}}\right)=1-g\left({i_{s+1}\over
m^{s+1}}+\dots+{i_{k}\over
m^{k}}\right)\leq\sum\limits_{j=1}^{k-s}2^{j}\left(1-g\left({{i_{s+j}\over
m^{s+j}}}\right)\right)}$
$\textstyle{\leq\sum\limits_{j=1}^{k-s}2^{j}2^{m}\left(1-g\left({{1\over
m^{s+j}}}\right)\right)\leq\sum\limits_{j=1}^{k-s}{C2^{m+j}\over
m^{s+j}}={C2^{m}\over m^{s}}\sum\limits_{j=1}^{k-s}\left({2\over
m}\right)^{j}}.$ (19)

In view of $m>2$, we find from (2) that the inequality

$\textstyle 1-g\left({i\over m^{k}}\right)\leq{C2^{m+1}\over(m-2)}{1\over
m^{s}}$ (20)

is valid. Obviously, (18) and (20) imply the continuity at zero of the
function $g(r)$ in the topology induced on $F$ by the topology of
$\mathbb{R}$, and hence, the uniform continuity of $g(r)$ in this topology.

It follows from (7) that the function $g(r)$ satisfies the equation

$g(r)=(g(p^{-1}r))^{k_{1}}\cdots(g(p^{-l}r))^{k_{l}},\ \ r\in F.$ (21)

We extend the function $g(r)$ by continuity from $F$ to a continuous positive
definite function on $\mathbb{R}$. We keep the notation $g$ for the extended
function. It follows from (21) that the function $g(s)$ satisfies the equation

$g(s)=(g(p^{-1}s))^{k_{1}}\cdots(g(p^{-l}s))^{k_{l}},\ \ s\in\mathbb{R}.$ (22)

By the generalized Pólya theorem, it follows from (8) and (22) that $g(s)$ is
the characteristic function of a symmetric Gaussian distribution on the real
line, and hence, we get the representation

$f(y)=\exp\\{-\sigma(y_{0})y^{2}\\},\ \ y=ry_{0},\ \ r\in F,$ (23)

where $\sigma(y_{0})\geq 0$.

When we obtained (23) we fixed an element $y_{0}\in Y$, $y_{0}\neq 0$, such
that $f(y_{0})>0$. If we fix another element $\tilde{y}\in Y$, $\tilde{y}\neq
0$, such that $f(\tilde{y})>0$, we will obtain the representation

$f(y)=\exp\\{-\sigma(\tilde{y})y^{2}\\},\ \ y=r\tilde{y},\ \ r\in F,$

where $\sigma(\tilde{y})\geq 0$. Since the subgroups $\\{y=ry_{0}:r\in F\\}$
and $\\{y=r\tilde{y}:r\in F\\}$ have a nonzero intersection, this implies that
$\sigma(y_{0})=\sigma(\tilde{y})=\sigma$. Put $E=\\{y\in Y:f(y)\neq 0\\}$.
Thus, we have the representation

$f(y)=\exp\\{-\sigma y^{2}\\},\ \ y\in E.$ (24)

We will verify that $E$ is a subgroup of $Y$. First, observe that when we got
(23) we proved that if $y\in E$, then $ry\in E$ for all $r\in F$. Take
$y_{1},y_{2}\in E$. It follows from (24) that $f(p^{-k}y_{j})\rightarrow 1$,
$j=1,2$, as $k\rightarrow\infty$, and then (14) implies that
$p^{-k}(y_{1}+y_{2})\in E$ for sufficiently large natural numbers $k$. Hence
$y_{1}+y_{2}\in E$.

Thus, we obtain the representation

$f(y)=\begin{cases}\exp\\{-\sigma y^{2}\\},&\text{\ if\ }\ y\in E,\\\
0,&\text{\ if\ }\ y\not\in E.\end{cases}$ (25)

Put $K=A(X,E)$. Let $\gamma$ be a Gaussian distribution on the group $X$ with
the characteristic function $\hat{\gamma}(y)=\exp\\{-\sigma y^{2}\\}$, $y\in
Y$. Since $E=A(Y,K)$, it follows from (2) and (25) that
$f(y)=\hat{\mu}(y)=\hat{\gamma}(y)\hat{m}_{K}(y)$, and hence
$\mu=\gamma*m_{K}$. Note that the subgroup $E$ has the property: if $py\in E$,
then $y\in E$. This implies that $f_{p}(K)=K$.

Get rid of the restriction that $\hat{\mu}(y)\geq 0$. Put $\nu=\mu*\bar{\mu}$.
Then $\hat{\nu}(y)=|\hat{\mu}(y)|^{2}\geq 0$ for all $y\in Y$, and the
characteristic function $\hat{\nu}(y)$ also satisfies equation (6). Then, as
has been proved above, the function $\hat{\nu}(y)$ is represented in the form
(25). On the one hand, the subgroup $E$ possesses the property: if $y\in E$,
then ${1\over p}y\in E$. Therefore $E$ is not isomorphic to the group
$\mathbb{Z}$. Taking into account that if a subgroup of the group of rational
numbers is not isomorphic to $\mathbb{Z}$, then it is isomorphic to a group of
the form (3), this implies that the group $E$ is topologically isomorphic to a
discrete additive group $H_{\text{\boldmath$b$}}$ of the rational numbers of
the form

$H_{\text{\boldmath$b$}}=\left\\{{m\over b_{0}b_{1}\cdots b_{n}}:\
n=0,1,\dots;\ m\in{\mathbb{Z}}\right\\}$

for some $b$=$(b_{0},b_{1},b_{2},\dots)$, where all $b_{j}\in{\mathbb{Z}}$,
$b_{j}>1$. On the other hand, since $E=A(Y,K)$, the group $E$ is topologically
isomorphic to the character group of the factor-group $X/K$. By the Pontryagin
duality theorem, the factor-group $X/K$ is topologically isomorphic to the
corresponding $b$-adic solenoid $\Sigma_{{\text{\boldmath$b$}}}$, and hence
the group $X/K$ contains no subgroup topologically isomorphic to the circle
group $\mathbb{T}$. Applying Lemma 2.5 to the group $X/K$, and taking into
account that each character of the subgroup $E$ can be written as $(x,y)$,
$x\in X$, $y\in E$, we get the statement of the theorem in the general case
from representation (25) for the function $\hat{\nu}(y)$. $\blacksquare$

######  Remark 2.6.

Let $\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid. Let $\gamma$ be
a Gaussian distribution on $\Sigma_{{\text{\boldmath$a$}}}$ with the
characteristic function of the form

$\hat{\gamma}(y)=\exp\\{-\sigma y^{2}\\},\ \ y\in H_{\text{\boldmath$a$}},$
(26)

where $\sigma\geq 0$. Let $\alpha_{j}$, $j=1,2,\dots,n$, $n\geq 2$, be
topological automorphisms of the group $\Sigma_{{\text{\boldmath$a$}}}$
satisfying the condition (4). It is obvious that the characteristic function
$\hat{\gamma}(y)$ satisfies equation (6).

Let $\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid with the property
that there is a prime number $p$ such that $\Sigma_{{\text{\boldmath$a$}}}$
contains no elements of order $p$. This implies that
$\Sigma_{{\text{\boldmath$a$}}}$ is a group with unique division by $p$. Let
$\alpha_{j}$, $j=1,2,\dots,n$, $n\geq 2$, be topological automorphisms of the
group $\Sigma_{{\text{\boldmath$a$}}}$ of the form $\alpha_{j}=\pm
f^{-1}_{p^{k_{j}}}$, where $k_{j}$ are natural numbers. Let $K$ be a compact
subgroup of $\Sigma_{{\text{\boldmath$a$}}}$ such that $f_{p}(K)=K$. Let
$\xi_{j}$ be independent identically distributed random variables with values
in $\Sigma_{{\text{\boldmath$a$}}}$ and distribution $m_{K}$. Note that the
condition $f_{p}(K)=K$ implies that if $py\in A(H_{\text{\boldmath$a$}},K)$,
then $y\in A(H_{\text{\boldmath$a$}},K)$. Taking this into account and using
(2) it is not difficult to verify that the characteristic function
$\hat{m}_{K}(y)$ also satisfies equation (6). In view of Lemma 2.4, it follows
from has been said the following statement.

Let $\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid with the property
that there is a prime number $p$ such that $\Sigma_{{\text{\boldmath$a$}}}$
contains no elements of order $p$. Let $\alpha_{j}$, $j=1,2,\dots,n$, $n\geq
2$, be topological automorphisms of $\Sigma_{{\text{\boldmath$a$}}}$ of the
form $\alpha_{j}=\pm f^{-1}_{p^{k_{j}}}$, where $k_{j}$ are natural numbers.
Assume that $\alpha_{j}$ satisfy the condition $(\ref{1})$. Let $\gamma$ be a
Gaussian distribution on $\Sigma_{{\text{\boldmath$a$}}}$ with the
characteristic function of the form $(\ref{12.02.1})$, and $K$ be a compact
subgroup of $\Sigma_{{\text{\boldmath$a$}}}$ such that $f_{p}(K)=K$. Put
$\mu=\gamma*m_{K}$. Let $\xi_{j}$ be independent identically distributed
random variables with values in $\Sigma_{{\text{\boldmath$a$}}}$ and
distribution $\mu$. Then $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are identically distributed.

Hence, we cannot narrow down the class of distributions in Theorem 2.1 which
is characterized by the property of equidistribution of $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$.

It turns out that Theorem 2.1 is false if the condition $(i)$ is not
satisfied. Namely, the following statement holds.

######  Proposition 2.7.

Let $X=\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid satisfying the
condition:

$(i)$ There are two prime numbers $p$ and $q$ such that the group $X$ contains
no elements of order $p$ and $q$.

Then there are topological automorphisms $\alpha_{j}$, $j=1,2,\dots,n$, $n\geq
2$, of the group $X$ satisfying the condition $(\ref{1})$, and independent
identically distributed random variables $\xi_{j}$ with values in $X$ and
distribution $\mu$ such that $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are identically distributed,
whereas $\mu$ can not be represented as a convolution $\mu=\gamma*m_{K}$,
where $\gamma\in\Gamma(X)$, and $K$ is a compact subgroup of the group $X$.

Theorem 2.1 and Proposition 2.7 imply the following statement.

######  Corollary 2.8.

Let $X=\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid. Let
$\alpha_{j}$, $j=1,2,\dots,n$, $n\geq 2$, be topological automorphisms of the
group $X$ satisfying the condition $(\ref{1})$. Let $\xi_{j}$ be independent
identically distributed random variables with values in $X$ and distribution
$\mu$. The equidistribution of $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ implies that $\mu=\gamma*m_{K}$,
where $\gamma\in\Gamma(X)$ and $K$ is a compact subgroup of the group $X$, if
and only if the condition $(i)$ of Theorem 2.1 is satisfied.

To prove Proposition 2.7 we need the following lemma ([16, (32.43)]).

######  Lemma 2.9.

Let $Y$ be an Abelian group, let $L$ be a subgroup of $Y$, and let $g(y)$ be a
positive definite function on $L$. If a function $f(y)$ on the group $Y$ is
represented in the form

$f(y)=\begin{cases}g(y),&\text{\ if\ }\ \ y\in L,\\\ 0,&\text{\ if\ }\ \
y\not\in L,\end{cases}$

then $f(y)$ is a positive definite function.

Proof of Proposition 2.7. Suppose for definiteness that $p<q$. In order not to
complicate the notation, we will identify $Y$ with $H_{\text{\boldmath$a$}}$.
The condition $(i)$ in Proposition 2.7 implies that $f_{p},f_{q}\in{\rm
Aut}(X)$, and hence, $f_{p},f_{q}\in{\rm Aut}(Y)$. Therefore both $X$ and $Y$
are groups with unique division by $p$ and by $q$. Denote by $H$ a subgroup of
$Y$ of the form

$H=\left\\{{m\over n}\in Y:n\ \text{is not divided by}\ p;\
m\in{\mathbb{Z}}\right\\}.$

Let $k$ be a natural number. Set

$H_{k}=\left\\{{m\over p^{k}n}\in Y:m,n\ \text{are not divided by}\
p\right\\}$

and

$L=H\cup H_{1}.$

It is obvious that $Y=H\cup\bigcup\limits_{k=1}^{\infty}H_{k}$ and $L$ is a
subgroup of $Y$. Denote by $G$ the character group of the group $L$, and put
$F=A(G,H)$. It is easy to see that $L/H\cong\mathbb{Z}(p)$, where
$\mathbb{Z}(p)$ is the group of residue classes modulo $p$. Since the
character group of the factor-group $L/H$ is topologically isomorphic to the
annihilator $A(G,H)$, we have $F\cong\mathbb{Z}(p)$. Take $0<c<1$. Let
$\omega$ be a distribution on the group $F$ of the form
$\omega=cE_{0}+(1-c)m_{F}$, where $E_{0}$ is the distribution concentrated at
zero. Consider $\omega$ as a distribution on the group $G$. Then (2) implies
that the characteristic function $\hat{\omega}(l)$, $l\in L$, is of the form

$\hat{\omega}(l)=\begin{cases}1,&\text{\ if\ }\ \ l\in H,\\\ c,&\text{\ if\ }\
\ l\in H_{1}.\end{cases}$

Obviously, $\hat{\omega}(l)$ is a positive definite function. Consider on the
group $Y$ the function

$f(y)=\begin{cases}\hat{\omega}(y),&\text{\ if\ }\ \ y\in L,\\\ 0,&\text{\ if\
}\ \ y\not\in L.\end{cases}$

By Lemma 2.9, $f(y)$ is a positive definite function. By the Bochner theorem,
there is a probability distribution $\mu$ on the group $X$ such that
$f(y)=\hat{\mu}(y)$. It is obvious that the distribution $\mu$ is not
represented as a convolution $\mu=\gamma*m_{K}$, where $\gamma\in\Gamma(X)$,
and $K$ is a compact subgroup of $X$.

Take natural numbers $l$ and $m$, where $l<m$, such that the remainders of the
division of the numbers $q^{2l}$ and $q^{2m}$ by $p^{2}$ are equal. Then
$q^{2m}-q^{2l}$ is divided by $p^{2}$. Hence, $q^{2m-2l}-1$ is also divided by
$p^{2}$, and the remainder of the division $q^{2m-2l}$ by $p^{2}$ is equal to
1. Put $a=m-l$. We have

$q^{2a}=p^{2}b+1.$ (27)

Put $\alpha_{1}=\dots=\alpha_{b}=f_{p}f^{-1}_{q^{a}}$,
$\alpha_{b+1}=f^{-1}_{q^{a}}$, $n=b+1$. In view of (27), the topological
automorphisms $\alpha_{j}$, $j=1,2,\dots,n$, of the group $X$ satisfy the
condition $(\ref{1})$.

Let $\xi_{j}$, $j=1,\dots n$, be independent identically distributed random
variables with values in $X$ and distribution $\mu$. Verify that the function
$f(y)$ satisfies the equation

$\textstyle f(y)=\left(f\left({p\over q^{a}}y\right)\right)^{b}f\left({1\over
q^{a}}y\right),\ \ y\in Y.$ (28)

Indeed, if $y\in H$, then ${p\over q^{a}}y,{1\over q^{a}}y\in H,$ and both
parts of equation (28) are equal to 1. If $y\in H_{1}$, then ${p\over
q^{a}}y\in H,$ ${1\over q^{a}}y\in H_{1},$ and both parts of equation (28) are
equal to $c$. If $y\in H_{k}$, $k\geq 2$, then ${1\over q^{a}}y\in H_{k},$ and
both parts of equation (28) are equal to zero. Thus, the function $f(y)$
satisfies equation (28), and by Lemma 2.4, $\xi_{j}$ and
$\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are identically distributed.
$\blacksquare$

######  Remark 2.10.

Let $X$ be a locally compact Abelian group and let $\nu\in{\rm M}^{1}(X)$. Put
$B=\\{y\in Y:\hat{\nu}(y)=1\\}$. It is well-known that $B$ is a closed
subgroup of $Y$ and the distribution $\nu$ is supported in the subgroup
$A(X,B)$. Therefore, in the notation of Proposition 2.7 the distribution $\mu$
constructed in the proof of Proposition 2.7 is supported in the annihilator
$A(X,H)$, i.e in a proper closed subgroup of the group $X$.

We can strengthen Proposition 2.7 as follows. Let
$X=\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid satisfying the
condition $(i)$ of Proposition 2.7. Let $\gamma$ be a non-degenerate Gaussian
distribution on $X$ with the characteristic function of the form (26). Set
$\lambda=\gamma*\mu$, where the distribution $\mu$ constructed in the proof of
Proposition 2.7. Then the characteristic function of the distribution
$\lambda$ is of the form

$\hat{\lambda}(y)=\begin{cases}\exp\\{-\sigma y^{2}\\},&\text{\ if\ }\ \ y\in
H,\\\ c\exp\\{-\sigma y^{2}\\},&\text{\ if\ }\ \ y\in H_{1},\\\ 0,&\text{\ if\
}\ \ y\not\in L.\end{cases}$

It is obvious that the distribution $\lambda$ can not be represented as a
convolution of a Gaussian distribution on $X$ and the Haar distribution of a
compact subgroup of the group $X$. Let $\xi_{j}$ be independent identically
distributed random variables with values in $X$ and distribution $\lambda$.
Arguing as in the proof of Proposition 2.7 and using Lemma 2.4, we are
convinced that $\xi_{j}$ and $\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$ are
identically distributed. As is easily seen, the distribution $\lambda$ is not
supported in any proper close subgroup of the group $X$.

######  Remark 2.11.

On the real line, both the Skitovich–Darmois theorem and the generalized Pólya
theorem characterize the Gaussian distribution. Corollary 2.8 gives a complete
description of $a$-adic solenoids on which the generalized Pólya theorem is
valid. It is interesting to note that, as follows from [12], there are no
$a$-adic solenoids on which the Skitovich–Darmois theorem holds. Namely, the
following statement takes place.

Let $X=\Sigma_{{\text{\boldmath$a$}}}$ be an $a$-adic solenoid. Then there are
topological automorphisms $\alpha_{j},\beta_{j}$ of the group $X$ and
independent random variables $\xi_{1}$ and $\xi_{2}$ with values in $X$ and
distributions $\mu_{1}$ and $\mu_{2}$ such that the linear forms
$L_{1}=\alpha_{1}\xi_{1}+\alpha_{2}\xi_{2}$ and
$L_{2}=\beta_{1}\xi_{1}+\beta_{2}\xi_{2}$ are independent, whereas $\mu_{j}$
can not be represented as convolutions $\mu_{j}=\gamma_{j}*m_{K_{j}}$, where
$\gamma_{j}\in\Gamma(X)$, and $K_{j}$ are compact subgroups of the group $X$,
$j=1,2$.

######  Remark 2.12.

As noted earlier each connected locally compact Abelian group of dimension 1
is topologically isomorphic to either the additive group of real numbers, or
the circle group, or an $a$-adic solenoid. The generalized Pólya theorem on
the additive group of real numbers characterizes Gaussian distributions
(Theorem A). The generalized Pólya theorem on $a$-adic solenoids
$\Sigma_{{\text{\boldmath$a$}}}$ satisfying the condition $(i)$ of Theorem 2.1
characterizes convolutions of Gaussian distributions on
$\Sigma_{{\text{\boldmath$a$}}}$ and Haar distributions on compact subgroups
of $\Sigma_{{\text{\boldmath$a$}}}$ (Theorem 2.1). Discuss the generalized
Pólya theorem on the circle group.

Let $\alpha_{j}$, $j=1,2,\dots,n$, $n\geq 2$, be topological automorphisms of
the circle group $\mathbb{T}$, and let $\xi_{j}$ be independent identically
distributed random variables with values in $\mathbb{T}$ and distribution
$\mu$. Consider the linear form $L=\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$.
Taking into account that $I$ and $-I$ are the only topological automorphisms
of $\mathbb{T}$, $L$ is of the form $L=\pm\xi_{1}+\dots+\pm\xi_{n}$. Assume
for definiteness that $L=\xi_{1}+\dots+\xi_{m}-\xi_{m+1}-\dots-\xi_{n}$. Set
$f(y)=\hat{\mu}(y)$. If $\xi_{j}$ and $L$ are identically distributed, then by
Lemma 2.4, the function $f(y)$ satisfies equation (6) which takes the form

$f(y)=(f(y))^{m}(f(-y))^{n-m},\ \ y\in\mathbb{Z}.$ (29)

Put $E=\\{y\in\mathbb{Z}:f(y)\neq 0\\}$. It follows from (29) that $|f(y)|=1$
for all $y\in E$. This implies that $E$ is a subgroup of $\mathbb{Z}$ and
there is an elements $x\in\mathbb{T}$ such that $f(y)=(x,y)$ for all $y\in E$.
Thus, we have

$f(y)=\begin{cases}(x,y),&\text{\ if\ }\ \ y\in E,\\\ 0,&\text{\ if\ }\ \
y\not\in E.\end{cases}$ (30)

Put $K=A(\mathbb{T},E)$. Then $E=A(\mathbb{Z},K)$. Taking into account (2), it
follows from (30) that $\mu$ is a shift by the element $x$ of the Haar
distribution $m_{K}$. We see that the generalized Pólya theorem on the circle
group $\mathbb{T}$ characterizes shifts of Haar distributions on compact
subgroups of $\mathbb{T}$.

Thus, we have completely studied the generalized Pólya theorem on connected
locally compact Abelian groups of dimension 1.

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