# On the Li–Zheng theorem

On the Li–Zheng theorem

Gennadiy Feldman

Abstract. By the well-known I. Kotlarski lemma, if $\xi_{1}$, $\xi_{2}$, and
$\xi_{3}$ are independent real-valued random variables with nonvanishing
characteristic functions, $L_{1}=\xi_{1}-\xi_{3}$ and $L_{2}=\xi_{2}-\xi_{3}$,
then the distribution of the random vector $(L_{1},L_{2})$ determines the
distributions of the random variables $\xi_{j}$ up to shift. Siran Li and
Xunjie Zheng generalized this result for the linear forms
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$ assuming that all $\xi_{j}$ have
first and second moments, $\xi_{2}$ and $\xi_{3}$ are identically distributed,
and $a_{j}$, $b_{j}$ satisfy some conditions. In the article, we give a
simpler proof of this theorem. In doing so, we also prove that the condition
of existence of moments can be omitted. Moreover, we prove an analogue of the
Li–Zheng theorem for independent random variables with values in the field of
$p$-adic numbers, in the field of integers modulo $p$, where $p\neq 2$, and in
the discrete field of rational numbers.

Mathematics Subject Classification. 39B52, 39A60, 60E05.

Keywords. Functional equation, Kotlarski lemma, Rao theorem, Field of $p$-adic
numbers.

##  1  Introduction

In the article [5] dedicated to the characterization of the gamma and the
Gaussian distribution I. Kotlarski proved the following lemma: Let $\xi_{1}$,
$\xi_{2}$, and $\xi_{3}$ be independent real-valued random variables with
nonvanishing characteristic functions, and let $L_{1}=\xi_{1}-\xi_{3}$ and
$L_{2}=\xi_{2}-\xi_{3}$. Then the distribution of the random vector
$(L_{1},L_{2})$ determines the distributions of the random variables $\xi_{j}$
up to shift. The characterization theorems proved in [5] remained known only
to mathematicians who deal with characterization problems of mathematical
statistics. At the same time, a number of studies that are far from
characterization problems are based on Kotlarski’s lemma. See, e.g., [6],
where numerous articles with references to Kotlarski’s lemma including in
economics literature, are mentioned. We especially pay attention to the
important article by C.R. Rao [7]. In particular, he considered $n$
independent real-valued random variables $\xi_{j}$ with nonvanishing
characteristic functions, the linear forms
$L_{1}=a_{1}\xi_{1}+\dots+a_{n}\xi_{n}$,
$L_{2}=b_{1}\xi_{1}+\dots+b_{n}\xi_{n}$ and proved that under some natural
conditions on the coefficients $a_{j}$, $b_{j}$ the characteristic function of
the random vector $(L_{1},L_{2})$ determines the characteristic functions of
the random variables $\xi_{j}$ up to factors of the form $\exp\\{P_{j}(y)\\}$,
where $P_{j}(y)$ is a polynomial of degree at most $n-2$. Kotlarski’s lemma
follows from this Rao theorem. Note also that some generalizations of Rao’s
theorem were studied in [1], see also [2, §15], for locally compact Abelian
groups. The proof of Rao’s theorem is based on the following statement on
solutions of a functional equation.

######  Lemma 1.1.

Let $a_{j}$, $b_{j}$, $j=1,2,\dots,n$, be nonzero real numbers such that
$a_{i}b_{j}\neq a_{j}b_{i}$ for all $i\neq j$. Consider the equation

$\sum_{j=1}^{n}\psi_{j}(a_{j}u+b_{j}v)=A(u)+B(v),\quad u,v\in\mathbb{R},$

where $\psi_{j}(y)$, $A(y)$, and $B(y)$ are continuous complex-valued
functions on $\mathbb{R}$. Then $\psi_{j}(y)$ are polynomial on $\mathbb{R}$
of degree at most $n$.

This lemma is well known. In fact, the lemma was first used, although it was
not explicitly formulated, in the works by Skitovich and Darmois, where the
Gaussian distribution on the real line is characterized by the independence of
two linear forms of $n$ independent random variables. Two different proofs of
the lemma and some generalizations can be found in [4, §1.5].

In article [6], Siran Li and Xunjie Zheng proved the following statement.

######  Li–Zheng theorem.

Let U and V be random variables with nonvanishing characteristic functions.
Assume that

$U=X+aZ_{1}+bZ_{2},\quad V=Y+cZ_{1}+dZ_{2},$

where $X$, $Y$, $Z_{1}$, and $Z_{2}$ are independent random variables with
well-defined first and second moments, $Z_{1}$ and $Z_{2}$ are identically
distributed, and $a$, $b$, $c$, and $d$ are nonzero real constants which are
known. Suppose that $ac\neq-bd$ and $(a,c)\neq(-b,-d)$. Then, the joint
distribution of $(U,V)$ uniquely determines the distributions of $X$, $Y$,
$Z_{1}$, and $Z_{2}$ up to a change of location.

Our article consists of three parts. In the first part, we give a simpler
proof of the Li–Zheng theorem and show that it essentially follows from Lemma
1.1. In doing so, we also prove that the condition of existence of moments of
the random variables $X$, $Y$, $Z_{1}$, and $Z_{2}$ can be omitted. In the
second part, we prove an analogue of the Li–Zheng theorem for independent
random variables with values in the field of $p$-adic numbers. In the third
part, we prove an analogue of the Li–Zheng theorem for independent random
variables with values in the field of integers modulo $p$, where $p\neq 2$,
and in the discrete field of rational numbers. For the proof of the
corresponding theorems we solve some functional equations on the character
group of the additive group of the field.

##  2  Real-valued random variables

Let us formulate the Li–Zheng theorem in more familiar for us notation. In
doing so, we omit the condition of existence of moments of independent random
variables.

######  Theorem 2.1.

Let $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent real-valued
random variables with nonvanishing characteristic functions. Let $a_{j}$,
$b_{j}$, $j=2,3$, be nonzero real numbers such that $a_{2}b_{2}\neq-
a_{3}b_{3}$ and $(a_{2},b_{2})\neq(-a_{3},-b_{3})$. Consider the linear forms
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. If the random variables $\xi_{2}$
and $\xi_{3}$ are identically distributed, then the distribution of the random
vector $(L_{1},L_{2})$ determines the distributions of the random variables
$\xi_{j}$, $j=1,2,3,4$, up to shift.

###### Proof.

1\. Let $\eta_{1}$, $\eta_{2}$, $\eta_{3}$, and $\eta_{4}$ be independent
real-valued random variables with nonvanishing characteristic functions.
Assume that $\eta_{2}$ and $\eta_{3}$ are identically distributed. Denote by
$\mu_{j}$ and $\nu_{j}$ the distributions of the random variables $\xi_{j}$
and $\eta_{j}$ and by $\hat{\mu}_{j}(y)$ and $\hat{\nu}_{j}(y)$ their
characteristic functions. Put $M_{1}=\eta_{1}+a_{2}\eta_{2}+a_{3}\eta_{3}$ and
$M_{2}=b_{2}\eta_{2}+b_{3}\eta_{3}+\eta_{4}$. Suppose that the distributions
of the random vectors $(L_{1},L_{2})$ and $(M_{1},M_{2})$ coincide. Taking
into account that the random variables $\xi_{j}$ are independent and
$\hat{\mu}_{j}(y)=\textbf{E}\left[e^{i\xi_{j}y}\right]$, the characteristic
function of the random vector $(L_{1},L_{2})$ can be represented in the form

$\textbf{E}\left[e^{i(L_{1}u+L_{2}v)}\right]=\textbf{E}\left[e^{i((\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3})u+(b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4})v)}\right]\\\
=\textbf{E}\left[e^{i\xi_{1}u}e^{i\xi_{2}(a_{2}u+b_{2}v)}e^{i\xi_{3}(a_{3}u+b_{3}v)}e^{i\xi_{4}v}\right]=\textbf{E}\left[e^{i\xi_{1}u}\right]\textbf{E}\left[e^{i\xi_{2}(a_{2}u+b_{2}v)}\right]\textbf{E}\left[e^{i\xi_{3}(a_{3}u+b_{3}v)}\right]\\\
\times\textbf{E}\left[e^{i\xi_{4}v}\right]=\hat{\mu}_{1}(u)\hat{\mu}_{2}(a_{2}u+b_{2}v)\hat{\mu}_{3}(a_{3}u+b_{3}v)\hat{\mu}_{4}(v),\quad
u,v\in\mathbb{R}.$ (1)

Analogically, the characteristic function of the random vector $(M_{1},M_{2})$
is of the form

$\textbf{E}\left[e^{i(M_{1}u+M_{2}v)}\right]=\hat{\nu}_{1}(u)\hat{\nu}_{2}(a_{2}u+b_{2}v)\hat{\nu}_{3}(a_{3}u+b_{3}v)\hat{\nu}_{4}(v),\quad
u,v\in\mathbb{R}.$ (2)

It follows from (1) and (2) that the random vectors $(L_{1},L_{2})$ and
$(M_{1},M_{2})$ have the same characteristic functions and hence they are
identically distributed if and only if the characteristic functions
$\hat{\mu}_{j}(y)$ and $\hat{\nu}_{j}(y)$ satisfy the equation

$\hat{\mu}_{1}(u)\hat{\mu}_{2}(a_{2}u+b_{2}v)\hat{\mu}_{3}(a_{3}u+b_{3}v)\hat{\mu}_{4}(v)=\hat{\nu}_{1}(u)\hat{\nu}_{2}(a_{2}u+b_{2}v)\hat{\nu}_{3}(a_{3}u+b_{3}v)\hat{\nu}_{4}(v),\quad
u,v\in\mathbb{R}.$ (3)

Set

$f_{j}(y)=\hat{\nu}_{j}(y)/\hat{\mu}_{j}(y),\quad\psi_{j}(y)=\ln
f_{j}(y),\quad j=1,2,3,4.$ (4)

Since the characteristic functions $\hat{\mu}_{j}(y)$ and $\hat{\nu}_{j}(y)$
do not vanish, (3) is equivalent to the fact that the functions $f_{j}(y)$
satisfy the equation

$f_{1}(u)f_{2}(a_{2}u+b_{2}v)f_{3}(a_{3}u+b_{3}v)f_{4}(v)=1,\quad
u,v\in\mathbb{R}.$ (5)

Note that we obtained equation (5) without assuming that the random variables
$\xi_{2}$ and $\xi_{3}$ and also $\eta_{2}$ and $\eta_{3}$ are identically
distributed.

It follows from (5) that the functions $\psi_{j}(y)$ satisfy the equation

$\psi_{1}(u)+\psi_{2}(a_{2}u+b_{2}v)+\psi_{3}(a_{3}u+b_{3}v)+\psi_{4}(v)=0,\quad
u,v\in\mathbb{R}.$ (6)

Rewrite this equation in the form

$\psi_{2}(a_{2}u+b_{2}v)+\psi_{3}(a_{3}u+b_{3}v)=A(u)+B(v),\quad
u,v\in\mathbb{R}.$ (7)

2\. First assume that $a_{2}b_{3}\neq a_{3}b_{2}$. Then by Lemma 1.1, the
functions $\psi_{2}(y)$ and $\psi_{3}(y)$ are polynomial of degree at most 2.
Taking into account that $\psi_{2}(0)=\psi_{3}(0)=0$, we have

$\psi_{j}(y)=\sigma_{j}y^{2}+\beta_{j}y,\quad y\in\mathbb{R},\ j=2,3,$ (8)

where $\sigma_{j}$, $\beta_{j}$ are complex numbers. Substituting (8) into (6)
and setting first $v=0$ and then $u=0$ in the obtained equation, we infer

$\psi_{j}(y)=\sigma_{j}y^{2}+\beta_{j}y,\quad y\in\mathbb{R},\ j=1,4,$ (9)

where $\sigma_{j}$, $\beta_{j}$ are complex numbers. Substitute (8) and (9)
into (6). We get from the received equation

$\sigma_{1}u^{2}+\sigma_{2}(a_{2}u+b_{2}v)^{2}+\sigma_{3}(a_{3}u+b_{3}v)^{2}+\sigma_{4}v^{2}=0,\quad
u,v\in\mathbb{R}.$ (10)

It follows from (10) that

$\sigma_{2}a_{2}b_{2}+\sigma_{3}a_{3}b_{3}=0.$ (11)

Since $\xi_{2}$ and $\xi_{3}$ are identically distributed and $\eta_{2}$ and
$\eta_{3}$ are also identically distributed, we have
$\hat{\mu}_{2}(y)=\hat{\mu}_{3}(y)$ and $\hat{\nu}_{2}(y)=\hat{\nu}_{3}(y)$,
$y\in\mathbb{R}$. Hence $\psi_{2}(y)=\psi_{3}(y)$, $y\in\mathbb{R}$. This
implies in particular that $\sigma_{2}=\sigma_{3}$. Inasmuch as
$a_{2}b_{2}+a_{3}b_{3}\neq 0$ by the conditions of the theorem, we get from
(11) that $\sigma_{2}=\sigma_{3}=0$. Then it follows from (10) that
$\sigma_{1}=\sigma_{4}=0$. Thus

$\psi_{j}(y)=\beta_{j}y,\quad y\in\mathbb{R},\ j=1,2,3,4.$ (12)

In view of (4), $\psi_{j}(-y)=\overline{\psi_{j}(y)}$ for all $y\in\mathbb{R}$
and (12) implies that $\beta_{j}=i\alpha_{j}$, where $\alpha_{j}$ are real
numbers. Hence $f_{j}(y)=e^{i\alpha_{j}y}$. Thus we proved that

$\hat{\nu}_{j}(y)=\hat{\mu}_{j}(y)e^{i\alpha_{j}y},\quad y\in\mathbb{R}.$

It follows from this that

$\nu_{j}=\mu_{j}*E_{\alpha_{j}},\quad j=1,2,3,4,$

where $E_{\alpha_{j}}$ is the degenerate distribution concentrated at the
point $\alpha_{j}$. So, if $a_{2}b_{3}\neq a_{3}b_{2}$, the theorem is proved.

3\. Assume now that $a_{2}b_{3}=a_{3}b_{2}$. In this case, we can not apply
Lemma 1.1 for solving equation (7), but equation (7) can be easily solved
directly.

Since $\xi_{2}$ and $\xi_{3}$ are identically distributed and $\eta_{2}$ and
$\eta_{3}$ are also identically distributed, we have
$\psi_{2}(y)=\psi_{3}(y)$, $y\in\mathbb{R}$. Put

$\psi(y)=\psi_{2}(y)=\psi_{3}(y).$ (13)

In view of $a_{2}b_{3}=a_{3}b_{2}$, set $c=b_{2}/a_{2}=b_{3}/a_{3}$. Inasmuch
as $a_{2}u+b_{2}v=a_{2}(u+cv)$ and $a_{3}u+b_{3}v=a_{3}(u+cv)$, it is easy to
see that equation (7) can be rewritten in the form

$\psi(a_{2}(u+cv))+\psi(a_{3}(u+cv))=\psi(a_{2}u)+\psi(a_{3}u)+\psi(a_{2}cv)+\psi(a_{3}cv),\quad
u,v\in\mathbb{R}.$ (14)

Let $a_{2}=a_{3}$. Then (14) implies that $\psi(y)$ is a homogeneous linear
function. If we consider this fact, it follows from (6) and (13) that
$\psi_{1}(y)$ and $\psi_{4}(y)$ are also homogeneous linear functions, i.e.,
(12) is fulfilled. As noted in the proof of the final part of item 2, the
statement of the theorem follows from this.

Let $a_{2}\neq a_{3}$. Put

$\varphi(y)=\psi(a_{2}y)+\psi(a_{3}y).$

It follows from (14) that the function $\varphi(y)$ satisfies the equation

$\varphi(u+v)=\varphi(u)+\varphi(v),\quad u,v\in\mathbb{R}.$

Hence there is a complex number $a$ such that $\varphi(y)=ay$ for all
$y\in\mathbb{R}$. Consider the function

$\gamma(y)=\psi(y)-by,\quad y\in\mathbb{R},$ (15)

where $b=\frac{a}{a_{2}+a_{3}}$. Then we have

$\gamma(a_{2}y)+\gamma(a_{3}y)=\psi(a_{2}y)-ba_{2}y+\psi(a_{3}y)-ba_{3}y=\varphi(y)-ay=0.$

This implies that

$\gamma(y)=-\gamma(ky),\quad y\in\mathbb{R},$ (16)

where $k=\frac{a_{2}}{a_{3}}$. Since $a_{2}b_{3}=a_{3}b_{2}$ and
$(a_{2},b_{2})\neq(-a_{3},-b_{3})$, we have $k\neq-1$. By the condition,
$k\neq 1$. For this reason $|k|\neq 1$. Suppose for definiteness that $|k|<1$.
We get from (16)

$\gamma(y)=(-1)^{n}\gamma(k^{n}y),\quad y\in\mathbb{R},\ n=1,2,\dots$ (17)

Obviously, $k^{n}y\rightarrow 0$ as $n\rightarrow\infty$ for all
$y\in\mathbb{R}$. Taking into account that $\gamma(y)$ is a continuous
function and $\gamma(0)=0$, it follows from (17) that $\gamma(y)=0$ for all
$y\in\mathbb{R}$. Hence (15) implies that $\psi(y)$ is a homogeneous linear
function. Then it follows from (6) that $\psi_{1}(y)$ and $\psi_{4}(y)$ are
also homogeneous linear functions, i.e., (12) is fulfilled. As noted above,
the statement of the theorem follows from this. The theorem is completely
proved. ∎

Let $a_{j}$, $b_{j}$, $j=2,3$, be nonzero real numbers. It is obvious that if
$a_{2}b_{3}\neq a_{3}b_{2}$, then $(a_{2},b_{2})\neq(-a_{3},-b_{3})$. Assume
that $a_{2}b_{3}=a_{3}b_{2}$. This implies that $a_{2}b_{2}\neq-a_{3}b_{3}$,
and the condition $(a_{2},b_{2})\neq(-a_{3},-b_{3})$ is equivalent to the
condition that either $|a_{2}|\neq|a_{3}|$ or $a_{2}=a_{3}$. Taking this into
account, Theorem 2.1 can be reformulated as follows (compare below with
Theorems 3.2, 4.1, and 4.3).

######  Theorem 2.2.

Let $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent random
variables with values in $\mathbb{R}$ with nonvanishing characteristic
functions. Suppose that the random variables $\xi_{2}$ and $\xi_{3}$ are
identically distributed. Let $a_{j}$, $b_{j}$, $j=2,3$, be nonzero real
numbers. Consider the linear forms $L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$
and $L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. Assume that one of the
following conditions holds:

  1. (I)

$a_{2}b_{3}\neq a_{3}b_{2}$ and $a_{2}b_{2}\neq-a_{3}b_{3}$;

  2. (II)

$a_{2}b_{3}=a_{3}b_{2}$ and $|a_{2}|\neq|a_{3}|$;

  3. (III)

$a_{2}b_{3}=a_{3}b_{2}$ and $a_{2}=a_{3}$.

Then the distribution of the random vector $(L_{1},L_{2})$ determines the
distributions of the random variables $\xi_{j}$, $j=1,2,3,4$, up to shift.

######  Remark 2.3.

Let us assume that in Theorem 2.2 the coefficients $a_{j}$ and $b_{j}$ satisfy
the condition

(IV) $a_{2}b_{3}=a_{3}b_{2}$ and $a_{2}=-a_{3}$.

We will verify that in this case the distribution of the random vector
$(L_{1},L_{2})$ uniquely determines the distributions of the random variables
$\xi_{1}$ and $\xi_{4}$, i.e., $\nu_{1}=\mu_{1}$ and $\nu_{4}=\mu_{4}$, but
need not necessarily determines the distribution of the random variables
$\xi_{2}$ and $\xi_{3}$ up to shift. Taking into account that
$|a_{2}|=|a_{3}|$ if and only if either $a_{2}=a_{3}$ or $a_{2}=-a_{3}$, it
means that we can not strengthen Theorem 2.2 and replace conditions (II) and
(III) in Theorem 2.2 by the condition $a_{2}b_{3}=a_{3}b_{2}$.

Since the random variables $\xi_{2}$ and $\xi_{3}$ are identically distributed
and $\eta_{2}$ and $\eta_{3}$ are also identically distributed, put
$f(y)=f_{2}(y)=f_{3}(y)$. In view of (IV), equation (5) takes the form

$f_{1}(u)|f(a_{2}(u+cv))|^{2}f_{4}(v)=1,\quad u,v\in\mathbb{R}.$ (18)

Set $l(y)=|f(a_{2}y)|^{2}$. It follows from (18) that the function $l(y)$
satisfies the equation

$l(u+v)=l(u)l(v),\quad u,v\in\mathbb{R}.$

Hence $l(y)=e^{\kappa y}$, where $\kappa\in\mathbb{R}$. Inasmuch as
$l(-y)=l(y)$, we have $\kappa=0$, i.e., $l(y)=1$ for all $y\in\mathbb{R}$.
Taking this into account, we get from equation (18) that $f_{1}(y)=f_{4}(y)=1$
for all $y\in\mathbb{R}$. Hence $\hat{\nu}_{1}(y)=\hat{\mu}_{1}(y)$ and
$\hat{\nu}_{4}(y)=\hat{\mu}_{4}(y)$ for all $y\in\mathbb{R}$. This implies
that $\nu_{1}=\mu_{1}$ and $\nu_{4}=\mu_{4}$.

Consider the distributions $\mu$ and $\nu$ with the characteristic functions

$\hat{\mu}(y)=\exp\\{(e^{iy}-1)\\},\quad\hat{\nu}(y)=\exp\\{(e^{-iy}-1)\\},\quad
y\in\mathbb{R}.$

Then we have

$|\hat{\mu}(y)|=|\hat{\nu}(y)|=\exp\\{\cos y-1\\},\quad y\in\mathbb{R}.$

This implies that

$|f(y)|=1,\quad y\in\mathbb{R}.$ (19)

It is obvious that $\nu$ is not a shift of $\mu$. Moreover, It is easy to see
that there is no a distribution $\lambda$ such that either $\nu=\mu*\lambda$
or $\mu=\nu*\lambda$.

Let $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent random
variables such that $\xi_{2}$ and $\xi_{3}$ are identically distributed.
Assume that the random variable $\xi_{j}$ has the distribution $\mu_{j}$,
$j=1,2,3,4$, where $\mu_{1}$ and $\mu_{4}$ are arbitrary distributions with
nonvanishing characteristic functions and $\mu_{2}=\mu_{3}=\mu$. Consider the
linear forms $L_{1}=\xi_{1}+a_{2}\xi_{2}-a_{2}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}-b_{2}\xi_{3}+\xi_{4}$. Let $\eta_{1}$, $\eta_{2}$,
$\eta_{3}$, and $\eta_{4}$ be independent random variables such that
$\eta_{2}$ and $\eta_{3}$ are identically distributed. Suppose that $\eta_{j}$
has the distribution $\nu_{j}$, $j=1,2,3,4$, where $\nu_{1}=\mu_{1}$,
$\nu_{2}=\nu_{3}=\nu$, and $\nu_{4}=\mu_{4}$. Put
$M_{1}=\eta_{1}+a_{2}\eta_{2}-a_{2}\eta_{3}$ and
$M_{2}=b_{2}\eta_{2}-b_{2}\eta_{3}+\eta_{4}$.

Taking into account that $f_{1}(y)=f_{4}(y)=1$ for all $y\in\mathbb{R}$ and
(19), we see that the functions $f_{1}(y)$, $f(y)$, and $f_{4}(y)$ satisfy
equation (18). Hence the random vectors $(L_{1},L_{2})$ and $(M_{1},M_{2})$
are identically distributed, while $\nu$ is not a shift of $\mu$.

##  3  Random variables with values in the field of $p$-adic numbers

Let $X$ be a locally compact Abelian group, $Y$ be its character group. Denote
by $(x,y)$ the value of a character $y\in Y$ at an element $x\in X$. Let $\mu$
be a distribution on $X$. Denote by

$\hat{\mu}(y)=\int_{X}(x,y)d\mu(x),\quad y\in Y,$ (20)

the characteristic function of the distribution $\mu$.

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

$\Delta_{h}f(y)=f(y+h)-f(y),\quad y\in Y.$

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

$\Delta_{h}^{n+1}f(y)=0$

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

We need the following well-known statement (for the proof see, e.g., [2,
Proposition 1.30]).

######  Lemma 3.1.

Let $Y$ be a locally compact Abelian group such that all its elements are
compact. Then any continuous polynomial on $Y$ is a constant.

Consider the field of $p$-adic numbers $\mathbb{Q}_{p}$. When we say the group
$\mathbb{Q}_{p}$, we mean the additive group of the field $\mathbb{Q}_{p}$.
The group $\mathbb{Q}_{p}$ is a locally compact Abelian group. Its character
group is topologically isomorphic to $\mathbb{Q}_{p}$ ([3, (25.1)]).
Multiplication by a nonzero element of $\mathbb{Q}_{p}$ is a topological
automorphism of the group $\mathbb{Q}_{p}$. Note that $(ax,y)=(x,ay)$ for all
$a,x,y\in\mathbb{Q}_{p}$. If $\mu$ is a distribution on $\mathbb{Q}_{p}$, the
characteristic function $\hat{\mu}(y)$ is defined by formula (20), where
$X=Y=\mathbb{Q}_{p}$. The group $\mathbb{Q}_{p}$ is totally disconnected and
consists of compact elements. Denote by $|\cdot|_{p}$ the norm in the field
$\mathbb{Q}_{p}$.

In this section, we prove the following analogue of the Li–Zheng theorem for
the field $\mathbb{Q}_{p}$.

######  Theorem 3.2.

Let $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent random
variables with values in the field $\mathbb{Q}_{p}$ with nonvanishing
characteristic functions. Let $a_{j}$, $b_{j}$, $j=2,3$, be nonzero elements
of $\mathbb{Q}_{p}$. Consider the linear forms
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. Assume that one of the following
conditions holds:

  1. (I)

$a_{2}b_{3}\neq a_{3}b_{2}$;

  2. (II)

$a_{2}b_{3}=a_{3}b_{2}$, $|a_{2}|_{p}\neq|a_{3}|_{p}$, and the random
variables $\xi_{2}$ and $\xi_{3}$ are identically distributed.

Then the distribution of the random vector $(L_{1},L_{2})$ determines the
distributions of the random variables $\xi_{j}$, $j=1,2,3,4$, up to shift.

###### Proof.

Consider independent random variables $\eta_{1}$, $\eta_{2}$, $\eta_{3}$, and
$\eta_{4}$ with values in the field $\mathbb{Q}_{p}$ with nonvanishing
characteristic functions. Put $M_{1}=\eta_{1}+a_{2}\eta_{2}+a_{3}\eta_{3}$ and
$M_{2}=b_{2}\eta_{2}+b_{3}\eta_{3}+\eta_{4}$. Suppose that the distributions
of the random vectors $(L_{1},L_{2})$ and $(M_{1},M_{2})$ coincide. Note that
if $\xi$ is a random variable with values in $\mathbb{Q}_{p}$ and distribution
$\mu$, then $\hat{\mu}(y)=\textbf{E}[(\xi,y)]$. Taking this into account and
the fact that $(ax,y)=(x,ay)$ for all $a,x,y\in\mathbb{Q}_{p}$, we can argue
as in item 1 of the proof of Theorem 2.1. Keeping the same notation, we arrive
at the equation

$f_{1}(u)f_{2}(a_{2}u+b_{2}v)f_{3}(a_{3}u+b_{3}v)f_{4}(v)=1,\quad
u,v\in\mathbb{Q}_{p}.$ (21)

1\. Assume that condition (I) holds. Since we do not suppose that $\xi_{2}$
and $\xi_{3}$ are identically distributed and $\eta_{2}$ and $\eta_{3}$ are
identically distributed, we can assume, without loss of generality, that
$L_{1}=\xi_{1}+\xi_{2}+\xi_{3}$ and $M_{1}=\eta_{1}+\eta_{2}+\eta_{3}$. Then
equation (21) takes the form

$f_{1}(u)f_{2}(u+b_{2}v)f_{3}(u+b_{3}v)f_{4}(v)=1,\quad u,v\in\mathbb{Q}_{p},$
(22)

and the condition $a_{2}b_{3}\neq a_{3}b_{2}$ is transform to the condition
$b_{2}\neq b_{3}$. The group $\mathbb{Q}_{p}$ is totally disconnected. For
this reason, as opposed to the case of the real line, we can not take the
logarithm of both sides of equation (22) and pass to the corresponding
additive equation.

To solve equation (22) we use a slightly different approach and split the
solution of equation (22) into two parts. First we prove that $|f_{j}(y)|=1$
for all $y\in\mathbb{Q}_{p}$, $j=2,3$. Then we prove that the functions
$f_{j}(y)$, $j=1,2,3,4$, are characters of the $\mathbb{Q}_{p}$. It is obvious
that the statement of the theorem follows from this.

Put

$\theta_{j}(y)=\ln|f_{j}(y)|,\quad j=1,2,3,4.$

It follows from (22) that the functions $\theta_{j}(y)$ satisfy the equation

$\theta_{1}(u)+\theta_{2}(u+b_{2}v)+\theta_{3}(u+b_{3}v)+\theta_{4}(v)=0,\quad
u,v\in\mathbb{Q}_{p},$

which can be written in the form

$\theta_{2}(u+b_{2}v)+\theta_{3}(u+b_{3}v)=C(u)+D(v),\quad
u,v\in\mathbb{Q}_{p}.$ (23)

For solving equation (23) we use the finite difference method. The reasoning
is standard and the same as in the case of the real line. We present it here
for completeness.

Let $g$ be an arbitrary element of the group $\mathbb{Q}_{p}$. Substitute
$u-{b_{3}}g$ for $u$ and $v+g$ for $v$ in equation (23). Subtracting (23) from
the resulting equation we get

$\Delta_{(b_{2}-b_{3})g}{\theta_{2}(u+b_{2}v)}=\Delta_{-{b_{3}}g}C(u)+\Delta_{g}D(v),\quad
u,v\in\mathbb{Q}_{p}.$ (24)

Let $h$ be an arbitrary element of the group $\mathbb{Q}_{p}$. Substitute
$u+h$ for $u$ in equation (24). Subtracting (24) from the resulting equation
we obtain

$\Delta_{h}\Delta_{(b_{2}-b_{3})g}{\theta_{2}(u+b_{2}v)}=\Delta_{h}\Delta_{-{b_{3}}g}C(u),\quad
u,v\in\mathbb{Q}_{p}.$ (25)

Let $k$ be an arbitrary element of the group $\mathbb{Q}_{p}$. Substitute
$v+k$ for $v$ in equation (25). Subtracting (25) from the resulting equation
we get

$\Delta_{b_{2}k}\Delta_{h}\Delta_{(b_{2}-b_{3})g}{\theta_{2}(u+b_{2}v)}=0,\quad
u,v\in\mathbb{Q}_{p}.$ (26)

Substituting $v=0$ in equation (26), we obtain

$\Delta_{b_{2}k}\Delta_{h}\Delta_{(b_{2}-b_{3})g}{\theta_{2}(u)}=0,\quad
u\in\mathbb{Q}_{p}.$

Since $b_{2}-b_{3}\neq 0$ and $g$, $h$ and $k$ are arbitrary elements of the
group $\mathbb{Q}_{p}$, we conclude that the function $\theta_{2}(y)$
satisfies the equation

$\Delta^{3}_{h}{\theta_{2}(y)}=0,\quad y,h\in\mathbb{Q}_{p},$

i.e., is a polynomial on $\mathbb{Q}_{p}$. Since the group $\mathbb{Q}_{p}$
consists of compact elements and the polynomial $\theta_{2}(y)$ is continuous,
by Lemma 3.1, $\theta_{2}(y)$ is a constant. In view of $\theta_{2}(0)=0$, we
have $\theta_{2}(y)=0$ for all $y\in\mathbb{Q}_{p}$. For the function
$\theta_{3}(y)$ we argue similarly excluding first the function
$\theta_{2}(y)$ from equation (23). Thus we proved that
$\theta_{2}(y)=\theta_{3}(y)=0$ and hence

$|f_{2}(y)|=|f_{3}(y)|=1,\quad y\in\mathbb{Q}_{p}.$

Let us prove that the functions $f_{j}(y)$, $j=1,2,3,4$, are characters of the
group $\mathbb{Q}_{p}$. Rewrite equation (22) in the form

$f_{2}(u+b_{2}v)f_{3}(u+b_{3}v)=S(u)T(v),\quad u,v\in\mathbb{Q}_{p}.$ (27)

To solve equation (27), we apply the method that was used to prove Theorem 3.1
in [1], see also [2, Theorem 15.8].

Let $g$ be an arbitrary element of the group $\mathbb{Q}_{p}$. Substitute
$u-b_{3}g$ for $u$ and $v+g$ for $v$ in equation (27). Dividing the resulting
equation by equation (27), we get

$\frac{f_{2}(u+b_{2}v-b_{3}g+b_{2}g)}{f_{2}(u+b_{2}v)}=\frac{S(u-b_{3}g)T(v+g)}{S(u)T(v)},\quad
u,v\in\mathbb{Q}_{p}.$ (28)

Let $h$ be an arbitrary element of the group $\mathbb{Q}_{p}$. Substitute
$u+h$ for $u$ in equation (28). Dividing the resulting equation by equation
(28), we receive

$\frac{f_{2}(u+b_{2}v-b_{3}g+b_{2}g+h)f_{2}(u+b_{2}v)}{f_{2}(u+b_{2}v+h)f_{2}(u+b_{2}v-b_{3}g+b_{2}g)}=\frac{S(u-b_{3}g+h)S(u)}{S(u+h)S(u-b_{3}g)},\quad
u,v\in\mathbb{Q}_{p}.$ (29)

Let $k$ be an arbitrary element of the group $\mathbb{Q}_{p}$. Substitute
$v+k$ for $v$ in equation (29). Dividing the resulting equation by equation
(29), we obtain

$\frac{f_{2}(u+b_{2}v-b_{3}g+b_{2}g+h+b_{2}k)f_{2}(u+b_{2}v+b_{2}k)}{f_{2}(u+b_{2}v+h+b_{2}k)f_{2}(u+b_{2}v-b_{3}g+b_{2}g+b_{2}k)}\\\
\times\frac{f_{2}(u+b_{2}v+h)f_{2}(u+b_{2}v-b_{3}g+b_{2}g)}{f_{2}(u+b_{2}v-b_{3}g+b_{2}g+h)f_{2}(u+b_{2}v)}=1,\quad
u,v\in\mathbb{Q}_{p}.$ (30)

Substitute in (30) $u=v=0$ and $g=-\frac{b_{2}k}{b_{2}-b_{3}}$. Then we get

$\frac{f^{2}_{2}(h)f_{2}(b_{2}k)f_{2}(-b_{2}k)}{f_{2}(h+b_{2}k)f_{2}(h-b_{2}k)}=1,\quad
h,k\in\mathbb{Q}_{p}.$ (31)

Taking into account that

$f_{2}(-y)=\overline{f_{2}(y)},\quad|f_{2}(y)|=1,\quad y\in\mathbb{Q}_{p},$
(32)

we have $f_{2}(b_{2}k)f_{2}(-b_{2}k)=1$ for all $k\in\mathbb{Q}_{p}$. Then it
follows from (31) that the function $f_{2}(y)$ satisfies the equation

$f^{2}_{2}(u)=f_{2}(u+v)f_{2}(u-v),\quad u,v\in\mathbb{Q}_{p}.$ (33)

In view of (32), we get from (33) that

$f^{2}_{2}(u+v)=f^{2}_{2}(u)f^{2}_{2}(v),\quad u,v\in\mathbb{Q}_{p},$ (34)

i.e., the function $f^{2}_{2}(y)$ is a character of the group
$\mathbb{Q}_{p}$. Substituting $u=v=y$ in (33), we obtain

$f^{2}_{2}(y)=f_{2}(2y),\quad y\in\mathbb{Q}_{p}.$ (35)

Taking into account that the mapping $y\rightarrow 2y$ is a topological
automorphism of the group $\mathbb{Q}_{p}$, it follows from (34) and (35) that
the function $f_{2}(y)$ satisfies the equation

$f_{2}(u+v)=f_{2}(u)f_{2}(v),\quad u,v\in\mathbb{Q}_{p},$

i.e., is a character of the group $\mathbb{Q}_{p}$. For the function
$f_{3}(y)$ we argue similarly. By the Pontryagin duality theorem, there are
elements $x_{2},x_{3}\in\mathbb{Q}_{p}$ such that

$f_{j}(y)=(x_{j},y),\quad y\in\mathbb{Q}_{p},\ j=2,3.$ (36)

Substituting (36) into (22) and putting first $v=0$ and then $u=0$ in the
obtained equation, we get that the functions $f_{1}(y)$ and $f_{4}(y)$ are
also characters of the group $\mathbb{Q}_{p}$.

Thus, we have proved that the characteristic functions $\hat{\mu}_{j}(y)$,
$j=1,2,3,4$, are determined up to multiplication by a character. Hence the
distributions $\mu_{j}$, $j=1,2,3,4$, are determined up to shift. The theorem
is proved if condition (I) is satisfied.

2\. Assume that condition (II) holds. Put

$c=b_{2}/a_{2}=b_{3}/a_{3},\quad
g(y)=f_{2}(a_{2}y)f_{3}(a_{3}y),\quad\tau(y)=\ln|g(y)|.$ (37)

2$a$. We prove in this item that the functions $f_{1}(y)$, $f_{4}(y)$, and
$g(y)$ are characters of the group $\mathbb{Q}_{p}$. In doing so, we do not
assume that $|a_{2}|_{p}\neq|a_{3}|_{p}$ and the random variables $\xi_{2}$
and $\xi_{3}$ are identically distributed.

Since $a_{2}u+b_{2}v=a_{2}(u+cv)$ and $a_{3}u+b_{3}v=a_{3}(u+cv)$, rewrite
equation (21) in the form

$f_{1}(u)g(u+cv)f_{4}(v)=1,\quad u,v\in\mathbb{Q}_{p}.$ (38)

Substituting first $v=0$ and then $u=0$ in (38), we get that the function
$g(y)$ satisfies the equation

$g(u+v)=g(u)g(v),\quad u,v\in\mathbb{Q}_{p}.$ (39)

It follows from (39) that the function $\tau(y)$ satisfies the equation

$\tau(u+v)=\tau(u)+\tau(v),\quad u,v\in\mathbb{Q}_{p}.$

Hence $\tau(y)$ is a continuous polynomial. Since the group $\mathbb{Q}_{p}$
consists of compact elements, by Lemma 3.1, $\tau(y)$ is a constant. In view
of $\tau(0)=0$, we have $\tau(y)=0$ and hence $|g(y)|=1$ for all
$y\in\mathbb{Q}_{p}$. Taking into account (39), this means that the function
$g(y)$ is a character of the group $\mathbb{Q}_{p}$. By the Pontryagin duality
theorem, there is an element $a\in\mathbb{Q}_{p}$ such that

$g(y)=(a,y),\quad y\in\mathbb{Q}_{p}.$ (40)

Hence

$g(u+cv)=(a,u+cv),\quad u,v\in\mathbb{Q}_{p}.$ (41)

Substituting (41) into (38), we find from the resulting equation that the
functions $f_{1}(y)$ and $f_{4}(y)$ are also characters of the group
$\mathbb{Q}_{p}$.

Note that in the proof we did not use the fact that
$|a_{2}|_{p}\neq|a_{3}|_{p}$ and the random variables $\xi_{2}$ and $\xi_{3}$
are identically distributed.

2$b$. Suppose that the random variables $\eta_{2}$ and $\eta_{3}$ are
identically distributed. Since the random variables $\xi_{2}$ and $\xi_{3}$
are also identically distributed, we have $f_{2}(y)=f_{3}(y)$. Set

$f(y)=f_{2}(y)=f_{3}(y).$

Then

$g(y)=f(a_{2}y)f(a_{3}y).$ (42)

We will prove now that the function $f(y)$ is a character of the group
$\mathbb{Q}_{p}$. Set

$\theta(y)=\ln|f(y)|.$

We have

$\tau(y)=\ln|g(y)|=\ln|f(a_{2}y)f(a_{3}y)|=\theta(a_{2}y)+\theta(a_{3}y),\quad
y\in\mathbb{Q}_{p}.$

Since $\tau(y)=0$ for all $y\in\mathbb{Q}_{p}$, it follows from this that

$\theta(a_{2}y)+\theta(a_{3}y)=0,\quad y\in\mathbb{Q}_{p}.$ (43)

Inasmuch as $|a_{2}|_{p}\neq|a_{3}|_{p}$, assume for definiteness that
$|a_{2}|_{p}<|a_{3}|_{p}$. Put $k=\frac{a_{2}}{a_{3}}$. Then $|k|_{p}<1$. We
obtain from (43) that $\theta(y)=-\theta(ky)$, and hence

$\theta(y)=(-1)^{n}\theta(k^{n}y),\quad y\in\mathbb{Q}_{p},\ n=1,2,\dots.$
(44)

It is obvious that $|k^{n}y|_{p}\rightarrow 0$ as $n\rightarrow\infty$ for all
$y\in\mathbb{Q}_{p}$. Since $\theta(y)$ is a continuous function and
$\theta(0)=0$, we obtain from (44) that $\theta(y)=0$ and hence $|f(y)|=1$ for
all $y\in\mathbb{Q}_{p}$.

Since $|a_{2}|_{p}\neq|a_{3}|_{p}$, we have $a_{2}+a_{3}\neq 0$. Put

$b=\frac{a}{a_{2}+a_{3}},\quad h(y)=f(y)(-b,y).$ (45)

It follows from (40), (42), and (45) that

$h(a_{2}y)h(a_{3}y)=f(a_{2}y)(-b,a_{2}y)f(a_{3}y)(-b,a_{3}y)\\\
=g(y)(-b(a_{2}+a_{3}),y)=g(y)(-a,y)=1,\quad y\in\mathbb{Q}_{p}.$

Hence $h(y)=h^{-1}(ky)$. This implies that

$h(y)=h^{(-1)^{n}}(k^{n}y),\quad y\in\mathbb{Q}_{p},\ n=1,2,\dots.$ (46)

We have $|k^{n}y|_{p}\rightarrow 0$ as $n\rightarrow\infty$ for all
$y\in\mathbb{Q}_{p}$. Inasmuch as $h(y)$ is a continuous function and
$h(0)=1$, it follows from (46) that $h(y)=1$ for all $y\in\mathbb{Q}_{p}$, and
(45) implies that

$f(y)=(b,y),\quad y\in\mathbb{Q}_{p}.$ (47)

Taking into account (47) and the fact that $f_{1}(y)$ and $f_{4}(y)$ are also
characters of the group $\mathbb{Q}_{p}$, we see that the characteristic
functions $\hat{\mu}_{j}(y)$ are determined up to multiplication by a
character. Hence the distributions $\mu_{j}$ are determined up to shift. The
theorem is proved if condition (II) is satisfied, and hence is completely
proved. ∎

######  Corollary 3.3.

Let $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent random
variables with values in the field $\mathbb{Q}_{p}$ with nonvanishing
characteristic functions. Let $a_{j}$, $b_{j}$, $j=2,3$, be nonzero elements
of $\mathbb{Q}_{p}$. Consider the linear forms
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. Then the distribution of the random
vector $(L_{1},L_{2})$ determines the distributions of the random variables
$\xi_{1}$ and $\xi_{4}$ up to shift.

###### Proof.

By Theorem 3.2, in the case, when $a_{2}b_{3}\neq a_{3}b_{2}$, the statement
of the corollary is true. If $a_{2}b_{3}=a_{3}b_{2}$, as has been proved in
item 2$a$ of the proof of Theorem 3.2 the functions $f_{1}(y)$ and $f_{4}(y)$
are characters of the group $\mathbb{Q}_{p}$. The statement of the corollary
follows from this. ∎

We can not omit the condition $|a_{2}|_{p}\neq|a_{3}|_{p}$ in condition (II)
of Theorem 3.2. Namely, the following statement is true.

######  Proposition 3.4.

Let $a_{j}$, $b_{j}$, $j=2,3$, be nonzero elements of the field
$\mathbb{Q}_{p}$ such that $a_{2}b_{3}=a_{3}b_{2}$. Assume that
$|a_{2}|_{p}=|a_{3}|_{p}$. Then there are independent random variables
$\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ with values in $\mathbb{Q}_{p}$
with nonvanishing characteristic functions such that the following is true:

  1. (I)

the random variables $\xi_{2}$ and $\xi_{3}$ are identically distributed;

  2. (II)

the distribution of the random vector $(L_{1},L_{2})$, where
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$, need not necessarily determine the
distribution of the random variables $\xi_{2}$ and $\xi_{3}$ up to shift.

###### Proof.

Let $G$ be a second countable compact Abelian group, $H$ be its character
group. Then $H$ is a countable discrete Abelian group. Let $\alpha(h)$ be a
real-valued nonvanishing function on the group $H$ satisfying the conditions:

  1. (i)

$\alpha(0)=1$;

  2. (ii)

$\alpha(-h)=\alpha(h)$ for all $h\in H$;

  3. (iii)

$\sum\limits_{h\in H}|\alpha(h)|<2$;

Consider on the group $G$ the function

$\rho(g)=\sum\limits_{h\in H}\alpha(h)\overline{(g,h)},\quad g\in G.$

It follows from (i)–(iii) that $\rho(g)$ is the nonnegative density with
respect to the Haar distribution on $G$ of a distribution $\mu$ on the group
$G$ with the characteristic function $\hat{\mu}(h)=\alpha(h)$.

Put

$\beta(h)=\begin{cases}1&\text{\ if\ }\ \ h=0,\\\ -\alpha(h)&\text{\ if\ }\ \
h\neq 0.\end{cases}$

Then $\beta(h)$ is a real-valued nonvanishing function on the group $H$.
Obviously, the function $\beta(h)$ also satisfies conditions (i)–(iii). Hence
there is a distribution $\nu$ on the group $G$ with the characteristic
function $\hat{\nu}(h)=\beta(h)$. It is easy to see that if $G$ is not
isomorphic to the additive group of the integers modulo 2, then $\nu$ is not a
shift of $\mu$.

Let $G=\mathbb{Z}_{p}$ be the ring of $p$-adic integers. Then $\mathbb{Z}_{p}$
is a compact subgroup of the group $\mathbb{Q}_{p}$. Let $\mu$ and $\nu$ be
the distributions on $\mathbb{Z}_{p}$ constructed above. Let $\xi_{1}$,
$\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent random variables with
values in $\mathbb{Z}_{p}$ such that $\xi_{2}$ and $\xi_{3}$ are identically
distributed. Assume that the random variable $\xi_{j}$ has the distribution
$\mu_{j}$, $j=1,2,3,4$, where $\mu_{1}$ and $\mu_{4}$ are arbitrary
distributions on $\mathbb{Z}_{p}$ with nonvanishing characteristic functions
and $\mu_{2}=\mu_{3}=\mu$. Consider the linear forms
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. Let $\eta_{1}$, $\eta_{2}$,
$\eta_{3}$, and $\eta_{4}$ be independent random variables with values in
$\mathbb{Z}_{p}$ such that $\eta_{2}$ and $\eta_{3}$ are identically
distributed. Suppose that $\eta_{j}$ has the distribution $\nu_{j}$,
$j=1,2,3,4$, where $\nu_{1}=\mu_{1}$, $\nu_{2}=\nu_{3}=\nu$, and
$\nu_{4}=\mu_{4}$. Put $M_{1}=\eta_{1}+a_{2}\eta_{2}+a_{2}\eta_{3}$ and
$M_{2}=b_{2}\eta_{2}+b_{2}\eta_{3}+\eta_{4}$.

Consider $\xi_{j}$ and $\eta_{j}$ as independent random variables with values
in the field $\mathbb{Q}_{p}$ and verify that the characteristic functions of
the random vectors $(L_{1},L_{2})$ and $(M_{1},M_{2})$ coincide. This is
equivalent to the fact that the functions
$f_{j}(y)=\hat{\nu}_{j}(y)/\hat{\mu}_{j}(y)$ satisfy equation (21), where
$f_{2}(y)=f_{3}(y)=f(y)$. Taking into account that $f_{1}(y)=f_{4}(y)=1$ for
all $y\in\mathbb{Q}_{p}$, equation (21) takes the form

$f(a_{2}(u+cv))f(a_{3}(u+cv))=1,\quad u,v\in\mathbb{Q}_{p},$

where $c=b_{2}/a_{2}=b_{3}/a_{3}$, or

$f(a_{2}y)f(a_{3}y)=1,\quad y\in\mathbb{Q}_{p}.$ (48)

We consider the distributions $\mu$ and $\nu$ as distributions on
$\mathbb{Q}_{p}$. It is easy to see that then the function $f(y)$ is of the
form

$f(y)=\begin{cases}1&\text{\ if\ }\ \ y\in p\mathbb{Z}_{p},\\\ -1&\text{\ if\
}\ \ y\notin p\mathbb{Z}_{p}.\end{cases}$ (49)

It follows from (49) and the condition $|a_{2}|_{p}=|a_{3}|_{p}$ that the
function $f(y)$ satisfies equation (48). Thus, the characteristic functions of
the random vectors $(L_{1},L_{2})$ and $(M_{1},M_{2})$ coincide. Hence the
random vectors $(L_{1},L_{2})$ and $(M_{1},M_{2})$ are identically
distributed, while $\nu$ is not a shift of $\mu$.

We note that if $|a_{2}|_{p}\neq|a_{3}|_{p}$, then the function $f(y)$ does
not satisfy equation (48), i.e., the random vectors $(L_{1},L_{2})$ and
$(M_{1},M_{2})$ are not identically distributed. ∎

The condition $|a_{2}|_{p}=|a_{3}|_{p}$ is an analogue for the field
$\mathbb{Q}_{p}$ of the condition $|a_{2}|=|a_{3}|$ for the field
$\mathbb{R}$. Comparing the statement of Theorem 2.2 in the case when
condition (III) holds and Proposition 3.4, we see that in the field
$\mathbb{Q}_{p}$ do not exist nonzero elements $a_{j}$, $b_{j}$, $j=2,3$, such
that $a_{2}b_{3}=a_{3}b_{2}$ and $|a_{2}|_{p}=|a_{3}|_{p}$ and the
distribution of the random vector $(L_{1},L_{2})$ determines the distributions
of the random variables $\xi_{2}$ and $\xi_{3}$ up to shift.

##  4  Random variables with values in the field of integers modulo $p$, where
$p\neq 2$, and in the discrete field of rational numbers

Let $p$ be a prime number and $\mathbb{Z}(p)$ be the field of integers modulo
$p$. When we say the group $\mathbb{Z}(p)$, we mean the additive group of the
field $\mathbb{Z}(p)$, i.e., the additive group of the integers modulo $p$.
The character group of the group $\mathbb{Z}(p)$ is isomorphic to
$\mathbb{Z}(p)$. Multiplication by a nonzero element of $\mathbb{Z}(p)$ is an
automorphism of the group $\mathbb{Z}(p)$. Note that $(ax,y)=(x,ay)$ for all
$a,x,y\in\mathbb{Z}(p)$. If $\mu$ is a distribution on $\mathbb{Z}(p)$, the
characteristic function $\hat{\mu}(y)$ is defined by formula (20), where
$X=Y=\mathbb{Z}(p)$. The group $\mathbb{Z}(p)$ is finite and hence compact.

In this section, we prove analogues of the Li–Zheng theorem for the field of
integers modulo $p$, where $p\neq 2$, and for the discrete field of rational
numbers $\mathbb{Q}$.

######  Theorem 4.1.

Consider the field $\mathbb{Z}(p)$, where $p\neq 2$. Let $\xi_{1}$, $\xi_{2}$,
$\xi_{3}$, and $\xi_{4}$ be independent random variables with values in
$\mathbb{Z}(p)$ with nonvanishing characteristic functions. Let $a_{j}$,
$b_{j}$, $j=2,3$, be nonzero elements of $\mathbb{Z}(p)$. Consider the linear
forms $L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. Then the following statements hold.

  1. 1.

If $a_{2}b_{3}\neq a_{3}b_{2}$, then the distribution of the random vector
$(L_{1},L_{2})$ determines the distributions of the random variables
$\xi_{j}$, $j=1,2,3,4$, up to shift.

  2. 2.

If $a_{2}b_{3}=a_{3}b_{2}$, then the distribution of the random vector
$(L_{1},L_{2})$ determines the distributions of the random variables $\xi_{1}$
and $\xi_{4}$ up to shift.

###### Proof.

1\. Assume that $a_{2}b_{3}\neq a_{3}b_{2}$. The proof of Theorem 3.2 in the
case, when condition (I) holds, is based only on the following properties of
the group $\mathbb{Q}_{p}$: the group $\mathbb{Q}_{p}$ consists of compact
elements and the mapping $y\rightarrow 2y$ is a topological automorphism of
the group $\mathbb{Q}_{p}$. Both of these properties are also valid for the
group $\mathbb{Z}(p)$, where $p\neq 2$. For this reason the proof remains
unchanged if, instead of the field $\mathbb{Q}_{p}$, we consider the filed
$\mathbb{Z}(p)$. Thus, statement 1 is valid.

2\. Assume that $a_{2}b_{3}=a_{3}b_{2}$. The reasoning carried out in item
2$a$ of the proof of Theorem 3.2 is based only on the fact that the group
$\mathbb{Q}_{p}$ consists of compact elements. For this reason the proof
remains unchanged if, instead of the field $\mathbb{Q}_{p}$, we consider the
filed $\mathbb{Z}(p)$. Thus, statement 2 is valid. ∎

Theorem 4.1 can not be strengthened. Namely, the following statement is true.

######  Proposition 4.2.

Let $a_{j}$, $b_{j}$, $j=2,3$, be nonzero elements of the field
$\mathbb{Z}(p)$, where $p\neq 2$, such that $a_{2}b_{3}=a_{3}b_{2}$. Then
there are independent random variables $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and
$\xi_{4}$ with values in $\mathbb{Z}(p)$ with nonvanishing characteristic
functions such that the following is true:

  1. (I)

the random variables $\xi_{2}$ and $\xi_{3}$ are identically distributed;

  2. (II)

the distribution of the random vector $(L_{1},L_{2})$, where
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$, need not necessarily determine the
distribution of the random variables $\xi_{2}$ and $\xi_{3}$ up to shift.

###### Proof.

To prove the proposition, we argue for the field $\mathbb{Z}(p)$ in the same
way as in Proposition 3.4 we argued for the field $\mathbb{Q}_{p}$ and keep
the same notation. The only difference is that instead of (49) the function
$f(y)$ is of the form

$f(y)=\begin{cases}1&\text{\ if\ }\ \ y=0,\\\ -1&\text{\ if\ }\ \ y\neq
0,\end{cases}$ (50)

because we construct the distributions $\mu$ and $\nu$ at once on
$\mathbb{Z}(p)$. The proposition will be proved if we verify that that the
function $f(y)$ satisfies the equation

$f(a_{2}y)f(a_{3}y)=1,\quad y\in\mathbb{Z}(p).$

In view of (50), it is obvious. ∎

Proposition 4.2 shows that, unlike Theorems 2.2 and 3.2, in the field
$\mathbb{Z}(p)$ do not exist nonzero elements $a_{j}$, $b_{j}$, $j=2,3$, such
that $a_{2}b_{3}=a_{3}b_{2}$ and the distribution of the random vector
$(L_{1},L_{2})$ determines the distributions of the random variables $\xi_{2}$
and up $\xi_{3}$ to shift.

Let $\mathbb{Q}$ be the field of rational numbers considering in the discrete
topology. When we say the group $\mathbb{Q}$, we mean the additive group of
the field $\mathbb{Q}$. The character group of the group $\mathbb{Q}$ is
topologically isomorphic to the $\bm{a}$-adic solenoid $\Sigma_{\bm{a}}$,
where ${\bm{a}}=(2,3,4,\dots)$ ([3, (25.4)]). The group $\Sigma_{\bm{a}}$ is
compact. Since the multiplication by any nonzero integer is an automorphism of
the group $\mathbb{Q}$, the multiplication by any nonzero integer is a
topological automorphism of the group $\Sigma_{\bm{a}}$. Therefore, the
multiplication by any nonzero rational number is well-defined and is also a
topological automorphism in the group $\Sigma_{\bm{a}}$. Note that
$(ax,y)=(x,ay)$ for all $a,x\in\mathbb{Q}$, $y\in\Sigma_{\bm{a}}$. If $\mu$ is
a distribution on the group $\mathbb{Q}$, the characteristic function
$\hat{\mu}(y)$ is defined by formula (20), where $X=\mathbb{Q}$,
$Y=\Sigma_{\bm{a}}$. The set $\mathbb{Q}$ is a countable subset of
$\mathbb{R}$. For this reason, if a random variable $\xi$ take values in
$\mathbb{Q}$ we can consider $\xi$ as a random variable with values in
$\mathbb{R}$. This implies, in particular, that Theorem 2.2 is valid for the
field $\mathbb{Q}$, i.e., when $\xi_{j}$ take values in $\mathbb{Q}$ and
$a_{j},b_{j}\in\mathbb{Q}$. However, the fact that random variables $\xi_{j}$
can be considered as random variables taking values in the discrete field
$\mathbb{Q}$ allows us to strengthen Theorem 2.2 for the field $\mathbb{Q}$.

######  Theorem 4.3.

Let $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent random
variables with values in the field $\mathbb{Q}$ with nonvanishing
characteristic functions. Let $a_{j}$, $b_{j}$, $j=2,3$, be nonzero elements
of $\mathbb{Q}$. Consider the linear forms
$L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. Assume that one of the following
conditions holds:

  1. (I)

$a_{2}b_{3}\neq a_{3}b_{2}$;

  2. (II)

$a_{2}b_{3}=a_{3}b_{2}$, $|a_{2}|\neq|a_{3}|$, and the random variables
$\xi_{2}$ and $\xi_{3}$ are identically distributed;

  3. (III)

$a_{2}b_{3}=a_{3}b_{2}$, $a_{2}=a_{3}$, and the random variables $\xi_{2}$ and
$\xi_{3}$ are identically distributed.

Then the distribution of the random vector $(L_{1},L_{2})$ determines the
distributions of the random variables $\xi_{j}$, $j=1,2,3,4$, up to shift.

###### Proof.

Assume that condition (I) holds. The proof of Theorem 3.2 in the case, when
condition (I) holds, is based only on the following properties of the group
$\mathbb{Q}_{p}$: the group $\mathbb{Q}_{p}$ consists of compact elements and
the mapping $y\rightarrow 2y$ is a topological automorphism of the group
$\mathbb{Q}_{p}$. Both of these properties are also valid for the group
$\Sigma_{\bm{a}}$, where ${\bm{a}}=(2,3,4,\dots)$. For this reason the proof
remains unchanged if, instead of the field $\mathbb{Q}_{p}$, we consider the
filed $\mathbb{Q}$.

Note that if $a_{2}b_{3}\neq a_{3}b_{2}$, then it follows from Theorem 2.2
that the random vector $(L_{1},L_{2})$ determines the distributions of the
random variables $\xi_{j}$, $j=1,2,3,4$, up to shift, only if additional
conditions are satisfied, namely $a_{2}b_{2}\neq-a_{3}b_{3}$ and the random
variables $\xi_{2}$ and $\xi_{3}$ are identically distributed.

In the case, when condition (II) or (III) is satisfied, the corresponding
statements follows from the corresponding statements of Theorem 2.2 for the
field $\mathbb{R}$. ∎

######  Remark 4.4.

The distributions $\mu$ and $\nu$ constructed in Remark 2.3 in fact are the
distributions on the group of integers. We can consider $\mu$ and $\nu$ as as
distributions on $\mathbb{Q}$. Hence Remark 2.3 is valid for the random
variables with values in the field $\mathbb{Q}$ and it shows that we can not
strengthen Theorem 4.3 and replace conditions (II) and (III) in Theorem 4.3 by
the condition $a_{2}b_{3}=a_{3}b_{2}$.

In Theorems 2.2, 3.2, 4.1, and 4.3 independent random variables take values in
a locally compact field, namely in $\mathbb{R}$, $\mathbb{Q}_{p}$,
$\mathbb{Z}(p)$, and $\mathbb{Q}$. In doing so, coefficients of the linear
forms are elements of the field. We can also study a more general problem,
when independent random variables take values in a locally compact Abelian
group $X$, and coefficients of the forms are continuous endomorphisms of $X$.
Taking this into account, we formulate the following problem.

Let $X$ be a second countable locally compact Abelian group. Let $\xi_{1}$,
$\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ be independent random variables with
values in $X$ with nonvanishing characteristic functions. Let $a_{j}$,
$b_{j}$, $j=2,3$, be continuous endomorphisms of the group $X$. Consider the
linear forms $L_{1}=\xi_{1}+a_{2}\xi_{2}+a_{3}\xi_{3}$ and
$L_{2}=b_{2}\xi_{2}+b_{3}\xi_{3}+\xi_{4}$. What are the conditions on $a_{j}$,
$b_{j}$, and $\xi_{j}$ to guarantee that the distribution of the random vector
$(L_{1},L_{2})$ determines the distributions of the random variables
$\xi_{j}$, $j=1,2,3,4$, up to shift?

It follows from the results of the article that these conditions depend on the
group $X$.

A more general problem can also be formulated.

Let us assume that we know the distribution of the random vector
$(L_{1},L_{2})$. How uniquely does this distribution determine the
distributions of the random variables $\xi_{j}$, $j=1,2,3,4$?

In connection with this problem, we note that in the case when $X$ is an
arbitrary $\bm{a}$-adic solenoid $\Sigma_{\bm{a}}$, it follows from Theorem
3.1 in [1], see also [2, Theorem 15.8], that if
$L_{1}=\xi_{1}+\xi_{2}+\xi_{3}$ and $b_{2}$, $b_{3}$, and $b_{2}-b_{3}$ are
topological automorphisms of the group $\Sigma_{\bm{a}}$, then the
distribution of the random vector $(L_{1},L_{2})$ determines the distributions
of the random variables $\xi_{j}$, $j=1,2,3,4$, up to convolution with a
Gaussian distribution on $\Sigma_{\bm{a}}$.

Acknowledgements

This article was written during the author’s stay at the Department of
Mathematics University of Toronto as a Visiting Professor. I am very grateful
to Ilia Binder for his invitation and support.

Funding The author declares that no funds, grants, or other support were
received during the preparation of this manuscript.

Data Availability Statement Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

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Department of Mathematics University of Toronto
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