On a generalisation of the Skitovich--Darmois theorem for several linear forms on Abelian groups

TL;DR

This paper extends Kagan's theorem, which characterizes Gaussian distributions via linear forms, to a broad class of locally compact Abelian groups, generalizing the Skitovich--Darmois theorem.

Contribution

It generalizes Kagan's theorem to various Abelian groups, broadening the scope of distribution characterization via linear forms.

Findings

Kagan's theorem holds on a wide class of Abelian groups

The characterization of Gaussian distributions extends beyond real numbers

The results unify and generalize classical distribution characterization theorems

Abstract

A.M. Kagan introduced a class of distributions $\mathcal{D}_{m, k}$ in $\mathbb{R}^m$ and proved that if the joint distribution of $m$ linear forms of $n$ independent random variables belongs to the class $\mathcal{D}_{m, m-1}$, then the random variables are Gaussian. A.M. Kagan's theorem implies, in particular, the well-known Skitovich--Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of $n$ independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan's theorem is valid.

Equations

$${\Delta }_h\psi (y)=\psi (y+h)-\psi (y).$$

$${\Delta }_h^{n+1}\psi (y)=0,y,h\in Y,$$

$$\hat{\mu }(y)={\hat{\mu }}_{\xi }(y)=\mathbb{E}[(\xi ,y)]={\int }_X(x,y)d{\mu }_{\xi }(x),y\in Y,$$

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

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

$${\hat{\mu }}_{\mathbf{\zeta }}(y_1,\dots ,y_m)=\mathbb{E}[(\mathbf{\zeta },(y_1,\dots ,y_m))]=\prod _{i_1,\dots ,i_k}{R}_{i_1,\dots i_k}(y_{i_1},\dots y_{i_k}),y_i\in Y,$$

$$G_i=\{({\alpha }_{1i}x,\dots ,{\alpha }_{mi}x)\in X^m:x\in X\}.$$

$$\overline{G_i}\cap \overline{G_l}=\{0\},i,l\in \{1,2,\dots ,n\},i\ne l.$$

$$\psi\big{(}a_{1}y_{1}+\dots+a_{m}y_{m}\big{)}=\sum_{j=1}^{m}r_{j}(y_{1},\dots,y_{j-1},y_{j+1},\dots,y_{m}),\quad y_{j}\in Y,$$

$$\Delta_{a_{1}h_{1}}\dots\Delta_{a_{m}h_{m}}\psi\big{(}a_{1}y_{1}+\dots+a_{m}y_{m}\big{)}=0,\quad y_{j}\in Y.$$

$$\Delta^{m}_{h}\psi\big{(}y\big{)}=0,\quad y,h\in Y,$$

$$\sum_{i=1}^{n}\psi_{i}\big{(}a_{1i}y_{1}+\dots+a_{mi}y_{m}\big{)}=\sum_{j=1}^{m}s_{j}(y_{1},\dots,y_{j-1},y_{j+1},\dots,y_{m}),\quad y_{j}\in Y,$$

$$a_{1n}{h}_{n1}+\cdots +a_{mn}{h}_{nm}=0.$$

$$\sum_{i=1}^{n-1}\Delta_{l_{in}}\psi_{i}\big{(}a_{1i}y_{1}+\dots+a_{mi}y_{m}\big{)}=\sum_{j=1}^{m}\Delta_{\bar{\bf l}_{jm}}s_{j}(y_{1},\dots,y_{j-1},y_{j+1},\dots,y_{m}),\quad y_{j}\in Y,$$

$$a_{1,n-1}{h}_{n-1,1}+\cdots +a_{m,n-1}{h}_{n-1,m}=0$$

$$\Delta_{l_{12}}\dots\Delta_{l_{1n}}\psi_{1}\big{(}a_{11}y_{1}+\dots+a_{m1}y_{m}\big{)}=\sum_{j=1}^{m}r_{j}(y_{1},\dots,y_{j-1},y_{j+1},\dots,y_{m}),\quad y_{j}\in Y,\quad y_{j}\in Y,$$

$$l_{1i}=a_{11}{h}_{i1}+\cdots +a_{m1}{h}_{im},i=2,\dots ,n,$$

$$a_{1i}{h}_{i1}+\cdots +a_{mi}{h}_{im}=0,i=2,\dots ,n,$$

$$g_i(y_1,\dots ,y_m)=a_{1i}{y}_1+\cdots +a_{mi}{y}_m,i=1,2,\dots ,n.$$

$$\Delta_{l_{12}}\dots\Delta_{l_{1n}}\psi_{1}\big{(}y\big{)}$$

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

$$\hat{\mu}_{\bf{\boldsymbol{L}}}(y_{1},\dots,y_{m})={\mathbb{E}}[({\boldsymbol{L}},(y_{1},\dots,y_{m}))]={\mathbb{E}}\bigg{[}\prod_{i=1}^{m}(\alpha_{j1}\xi_{1}+\cdots+\alpha_{jn}\xi_{n},y_{j})\bigg{]}$$

$$={\mathbb{E}}\bigg{[}\prod_{i=1}^{n}(\xi_{i},a_{1i}y_{1}+\dots+a_{mi}y_{m})\bigg{]}=\prod_{i=1}^{n}\hat{\mu}_{i}(a_{1i}y_{1}+\dots+a_{mi}y_{m}),\quad y_{j}\in Y.$$

$$\prod _{i=1}^n{\hat{\mu }}_i(a_{1i}{y}_1+\cdots +a_{mi}{y}_m)=\prod _{j=1}^m{R}_j(y_1,\dots ,y_{j-1},y_{j+1},\dots ,y_m),y_j\in Y,$$

$$\bigcap _{p,k=1}^m\mathrm{Ker}({\alpha }_{ki}^{-1}{\alpha }_{kl}-{\alpha }_{pi}^{-1}{\alpha }_{pl})=\{0\}.$$

$$\mathrm{Ker}({\alpha }_i-{\alpha }_l)=\{0\}$$

$${\phi }_i(y)={\psi }_i(y),i=1,2,\dots ,q-1,{\phi }_q(y)={\psi }_q(y)+{\psi }_q(c_{q+1}y)+\cdots +{\psi }_q(c_{n}y).$$

$$\sum_{i=1}^{q}\varphi_{i}\big{(}\alpha_{1i}y_{1}+\dots+\alpha_{mi}y_{m}\big{)}=\sum_{j=1}^{m}s_{j}(y_{1},\dots,y_{j-1},y_{j+1},\dots,y_{m}),\quad y_{j}\in Y.$$

$${\hat{\mu }}_{\mathbf{\xi }}(y_1,\dots ,y_n)=\mathbb{E}[({\xi }_1,y_1)\cdots ({\xi }_n,y_n)]=$$

$$=\bigg{(}\prod_{j=1}^{n}\hat{\mu}_{\xi_{j}}(y_{j})\bigg{)}\exp\{q(y_{1},\dots,y_{n})\},\quad y_{j}\in Y,$$

Full text

[TABLE]

Assume that the following condition hold:

[TABLE]

Let $\xi_{i}$, $i=1,2,\dots,n$, be independent random variables with values in the group $X$ and distributions $\mu_{i}$ with non vanishing characteristic functions. Consider the linear forms $L_{j}=\alpha_{j1}\xi_{1}+\cdots+\alpha_{jn}\xi_{n}$, $j=1,2,\dots,m$. If the distribution of the random vector $\boldsymbol{L}$=$(L_{1},\dots,L_{m})$ belongs to the class $\mathcal{D}_{m,m-1}$, then all $\mu_{i}$ are Gaussian distributions.

For ease of reference we formulate the following property of adjoint homomorphisms as a lemma.

[TABLE]

where $r_{j}$ are arbitrary functions. Then $\psi(y)$ is a polynomial on $Y$.

Proof. We use the finite difference method. Let $h_{m}$ be an arbitrary element of the group $Y$. Substitute $y_{m}+h_{m}$ for $y_{m}$ in Equation (3). Subtracting Equation (3) from the resulting equation we get an equation where the right-hand side does not contain the function $r_{m}$. By repeating this operation, we consistently exclude all functions $r_{j}$ from the right-hand side of the resulting equations. After $m-1$ steps we get

[TABLE]

By Lemma 1, the subgroup ${a_{j}(Y)}$ is dense in $Y$. Since $h_{j}$ are arbitrary elements of the group $Y$, it follows from (4) that the function $\psi(y)$ satisfies the equation

[TABLE]

i.e. $\psi(y)$ is a polynomial. $\Box$

[TABLE]

where $s_{j}$ are arbitrary functions. Then all $\psi_{i}(y)$ are polynomial on $Y$.

Proof. We use the finite difference method. Let $h_{nj}$, $j=1,2,\dots,m$, be elements of the group $Y$ such that

[TABLE]

Substitute $y_{j}+h_{nj}$ for $y_{j}$ for all $j$ in Equation (5) and subtract Equation (5) from the resulting equation. We obtain

[TABLE]

where $l_{in}=a_{1i}h_{n1}+\dots+a_{mi}h_{nm}$, $i=1,2,\dots,n-1$, ${\bf{\bar{l}}}_{jm}=(h_{n1},\dots,h_{n,j-1},h_{n,j+1},\dots,h_{nm})$, $j=1,2,\dots,m$. The left-hand side of Equation (6) no longer contains the function $\psi_{n}$. In the second step taking elements $h_{n-1,j}$ in such a manner that the equality

[TABLE]

holds and reasoning similarly, we exclude the function $\psi_{n-1}$ from the left-hand side of Equation (6). By excluding successively the functions $\psi_{i}$ from the left-hand side of Equation (6), after $n-1$ steps we come to an equation of the form

[TABLE]

where

[TABLE]

and $r_{j}$ are some functions. It should be noted that when we got Equation (7) we chose elements $h_{ij}$ at every step in such a manner that the equalities

[TABLE]

were fulfilled. Consider the subgroups $G_{i}=\{(\alpha_{1i}x,\dots,\alpha_{mi}x)\in X^{m}:x\in X\}$, $i=1,2,\dots,n,$ in $X^{m}$. Denote by $g_{i}$ the continuous homomorphism $g_{i}:Y^{m}\mapsto Y$ of the form

[TABLE]

The adjoint to $g_{i}$ homomorphism $\tilde{g}_{i}:X\mapsto X^{m}$ is of the form $\tilde{g}_{i}(x)=(\alpha_{1i}x,\dots,\alpha_{mi}x)\in G_{i}$. Put $H_{i}={\rm Ker}\ g_{i}$. Denote by $p_{i}$ the natural embedding $p_{i}:H_{i}\mapsto Y^{m}$, and by $g_{li}$ the restriction of $g_{l}$ to $H_{i}$. We have $g_{li}=g_{l}p_{i}$. Consider the annihilator $A(X^{m},H_{i})$. It is easy to see that $A(X^{m},H_{i})=\overline{G_{i}}.$ The character group of the subgroup $H_{i}$ is topologically isomorphic to the factor-group $X^{m}/A(X^{m},H_{i})$, and the adjoint to $p_{i}$ homomorphism $\tilde{p}_{i}$ is a factor mapping $\tilde{p}_{i}:X^{m}\mapsto X/A(X^{m},H_{i})$. We note that $\tilde{g}_{li}=\tilde{p}_{i}\tilde{g}_{l}$. Take $x\in{\rm Ker}\ \tilde{g}_{li}$. Then $\tilde{g}_{l}(x)=(\alpha_{1l}x,\dots,\alpha_{ml}x)\in{\rm Ker}\ \tilde{p}_{i}=A(X^{m},H_{i})=\overline{G_{i}}$. It follows from (2) that $x=0$ for $i\neq l$. Thus, ${\rm Ker}\ \tilde{g}_{li}=\{0\}$. By Lemma 1, the subgroup $g_{li}(H_{i})$ is dense in $Y$.

Let us return to Equation (7). Since $\alpha_{j1}$, $j=1,2,\dots,m$, are continuous monomorphisms of the group $X$, by Lemma 2, the function

[TABLE]

is a polynomial on $Y$. Taking into account that the subgroups $g_{li}(H_{i})$, $i\neq l$, are dense in $Y$ and (8), we can suppose that in (10) $l_{12}=\dots=l_{1n}=h,$ where $h$ is an arbitrary element of the group $Y$, i.e. the function $\Delta_{h}^{n-1}\psi_{1}(y)$ is a polynomial on $Y$. Hence, the function $\psi_{1}(y)$ is also a polynomial on $Y$. For the functions $\psi_{j}(y)$, $j=2,3,\dots,n$ we reason similarly. $\Box$

We also need group analogues of the classical Cramér and Marcinkiewicz theorems. We will formulate them as lemmas.

[TABLE]

where $\psi(y)$ is a continuous polynomial. Then $\mu\in\Gamma(X)$.

Proof of Theorem 1. Let $Y$ be a character group of the group $X$. Let $\hat{\mu}_{\boldsymbol{L}}(y_{1},\dots,y_{m})$ be the characteristic function of the random vector $\boldsymbol{L}$. Put $a_{pi}=\tilde{\alpha}_{pi}$, $p=1,2,\dots,m$, $i=1,2,\dots,n$. Taking into account that the random variables $\xi_{i}$ are independent, the characteristic function $\hat{\mu}_{\boldsymbol{L}}(y_{1},\dots,y_{m})$ is of the form

[TABLE]

[TABLE]

Taking into account that the distribution of the random vector $\boldsymbol{L}$ belongs to the class $\mathcal{D}_{m,m-1}$, we may write

[TABLE]

where $R_{j}(y_{1},\dots,y_{j-1},y_{j+1},\dots,y_{m})$ are continuous functions such that $R_{j}(0,\dots,0)=1$.

Put $\nu_{i}=\mu_{i}*\bar{\mu}_{i}$. Then we have $\hat{\nu}_{i}(y)=|\mu_{i}(y)|^{2}>0$, the characteristic functions of the distributions $\hat{\nu}_{i}(y)$ also satisfy Equation (11), and all the factors in the left-hand side and right-hand side of Equation (11) are greater then zero. If we prove that all $\nu_{i}\in\Gamma(X),$ it follows from this by Lemma 4, that all $\mu_{i}\in\Gamma(X)$ too. Thus, we may assume from the beginning that all factors in the left-hand side and right-hand side of Equation (11) are greater then zero.

Put $\psi_{i}(y)=\log\hat{\mu}_{i}(y)$, $i=1,2,\dots,n$, $s_{j}=\log R_{j}$, $j=1,2,\dots,m$. It follows from (11) that the functions $\psi_{i}$ and $s_{j}$ satisfy Equation (5). Then by Lemma 3, $\psi_{i}(y)$ are polynomial on $Y$. Applying Lemma 5, we obtain that $\mu_{i}\in\Gamma(X)$, $i=1,2\dots,n$. $\Box$

3. Corollaries and generalizations of the main theorem

Assume that $X$ is a locally compact Abelian torsion free group. Then $X$ does not contain a subgroup topologically isomorphic to the circle group $\mathbb{T}$, and multiplication by a nonzero integer is a continuous monomorphism of the group $X$. Hence, Theorem 1 implies the following statement (compare with [17]).

Suppose that in Theorem 1 $\alpha_{ji}$, $j=1,2,\dots,m$, $i=1,2,\dots,n$, are topological automorphisms of the group $X$. Then $G_{i}$ are closed subgroups of the group $X^{m}$, and condition (2), as easy to see, is equivalent to the following condition:

For any fixed $i,l\in\{1,2,\dots,n\}$, $i\neq l$, the following equality holds:

[TABLE]

We note that if $m=2$, and $L_{1}=\xi_{1}+\dots+\xi_{n}$ and $L_{2}=\alpha_{1}\xi_{1}+\dots+\alpha_{n}\xi_{n}$, where $\alpha_{i}$ are topological automorphisms of the group $X$, then condition (12) takes the form:

[TABLE]

for all $i,l\in\{1,2,\dots,n\}$, $i\neq l$.

Let $X$ be a locally compact Abelian divisible torsion free group. We recall that an Abelian group $G$ is divisible if, for every natural $n$ and every $g\in G$, there exists $x\in G$ such that $nx=g$. An important example of a locally compact Abelian divisible torsion free group is a group of the form \mathbb{R}^{n}\times{\Sigma^{\mathfrak{n}}_{\text{\boldmatha}}}, where \Sigma_{\text{\boldmatha}} is an $a$-adic solenoid (a compact Abelian group such that its character group is topologically isomorphic to the group of rational numbers $\mathbb{Q}$ considering the discrete topology), ${\mathfrak{n}}$ is a cardinal number. Any connected locally compact Abelian divisible torsion free group is topologically isomorphic to a group of this type. For the structure theorem for an arbitrary locally compact Abelian divisible torsion free group see [13, (25.33)]. Obviously, for these groups $X$ multiplication by any nonzero integer, and hence, multiplication by any nonzero rational number, is a topological automorphism. Therefore, we may consider linear forms of independent random variables taking values in $X$ with rational coefficients. Corollary 1 for such groups can be significantly improved. Namely, condition (2) or the equivalent condition (12), can be omitted.

Proof. It is easy to see that the character group of a locally compact Abelian divisible torsion free group is also a locally compact Abelian divisible torsion free group. The proof of Theorem 1 is based on Lemma 3 and group analogues of theorems by Cramér and Marcinkiewicz (Lemmas 4 and 5). Since $X$ does not contain a subgroup topologically isomorphic to the circle group $\mathbb{T}$, Lemmas 4 and 5 hold for the group $X$. We retain the notation used in the proof of the Theorem 1. Reasoning as in the proof of Theorem 1, we come to Equation (5).

Consider integer-valued vectors ${\boldsymbol{\alpha}}_{i}=(\alpha_{1i},\dots,\alpha_{mi})$, $i=1,2,\dots,n$. Since multiplication by any nonzero rational number is a topological automorphism of the groups $X$ and $Y$, it is easy to see that condition (2) or equivalent condition (12), are not fulfilled if and only if the vectors ${\boldsymbol{\alpha}}_{i}$ and ${\boldsymbol{\alpha}}_{l}$ are collinear for some $i\neq l$. We assume for simplicity that we have the only subset of collinear vectors, and they are the vectors ${\boldsymbol{\alpha}}_{q},\dots,{\boldsymbol{\alpha}}_{n}$, and the vectors ${\boldsymbol{\alpha}}_{1},\dots,{\boldsymbol{\alpha}}_{q}$ are not collinear. The general case may be considered similarly. Let ${\boldsymbol{\alpha}}_{i}=c_{i}{\boldsymbol{\ \alpha}}_{q}$, $i=q+1,q+2,\dots,n$, where $c_{i}$ are nonzero rational numbers. Put

[TABLE]

Equation (5) with this notation takes the form

[TABLE]

The non-collinearity of the vectors ${\boldsymbol{\alpha}}_{i}$, $i=1,2,\dots,q$, implies that the topological automorphisms $\alpha_{pi}$, $p=1,2,\dots,m$, $i=1,2,\dots,q$, already satisfy condition (2) or equivalent condition (12). Now, the conditions of Lemma 3 are satisfied. By Lemma 3, the functions $\varphi_{i}(y)$, $i=1,2,\dots,q$, are polynomial on $Y$. Applying Lemma 5, we obtain that $\mu_{i}\in\Gamma(X)$, $i=1,2,\dots,q-1$. Moreover, $\exp\{\varphi_{q}(y)\}$ is the characteristic function of a Gaussian distribution. Taking into account that multiplication by $c_{i}$, $i=q+1,q+2,\dots,n$, are topological automorphisms of the group $Y$, and applying Lemma 4, we get that $\mu_{i}\in\Gamma(X)$, $i=q,q+1,\dots,n$.

It should be noted that if in Theorem 2 $m=2$, i.e. we have two independent linear forms $L_{1}$ and $L_{2}$ of $n$ independent random variables $\xi_{i}$, and the coefficients of the linear forms $\alpha_{ji}$, $j=1,2,\dots,m$, $i=1,2,\dots,n$, are nonzero integers, then, as it was proved in [4], the statement of Theorem 2 remains true if we omit the condition that $X$ is a divisible group. $\Box$

In the article [16] A. M. Kagan and G. J. Székely introduced a notion of $Q$-independence and proved, in particular, that the Skitovich–Darmois theorem holds true if instead of independence $Q$-independence is considered. Then in the article [10] a notion of $Q$-independence for random variables with values in a locally compact Abelian group was introduced. In [10] it was proved, in particular, that a group analogue of the Skitovich–Darmois theorem (see [7]) remains 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 random vector $\boldsymbol{\xi}$=$(\xi_{1},\dots,\xi_{n})$ is represented in the form

[TABLE]

[TABLE]

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

We will prove now that, exactly as in the case of the Skitovich–Darmois theorem, Theorem 1 holds true if we change the condition of independence of the random variables $\xi_{i}$ for $Q$-independence. The following statement holds.

Proof. We retain the notation used in the proof of Theorem 1, and reason as in the proof of Theorem 1. Taking into account the $Q$-independence of random variables $\xi_{i}$, the characteristic function of the random vector $\boldsymbol{L}$ is of the form

[TABLE]

[TABLE]

where $q(y_{1},\dots,y_{n})$ is a continuous polynomial on the group $Y^{n}$, and instead of Equation (11) we get the equation

[TABLE]

[TABLE]

where $p(y_{1},\dots,y_{m})$ is a continuous polynomial on the group $Y^{m}$. Following the proof of Theorem 1, we want to conclude from (S0.Ex19) that all distributions $\mu_{i}$ are Gaussian. To do this we need to know that the statements of Lemma 2 and 3 remain true in the case when we add a polynomial to the right-hand side of Equations (3) and (5). It is easy to see that it is true. The proofs of the lemmas are almost the same. The only difference is that the degrees of polynomials $\psi(y)$ and $\psi_{i}(y)$ are changed. $\Box$

Theorem 3 and the reasoning used in the proof of Theorem 2 imply the following analogue of Theorem 2 for Q-independent random variables.

Considering this we will formulate the following question. Does Theorem 1 hold true if we omit condition (2) or (12) in the case when the coefficients $\alpha_{ji}$ are topological automorphisms of the group $X$? **

Consider a more complicated example. Let $X=\mathbb{T}^{2}$. The main theorem proved in [18] implies the complete description of topological automorphisms $\alpha$ of the group $X$ which possess the following property: there exist independent non-Gaussian random variables $\xi_{1}$ and $\xi_{1}$ with values in the group $X$ with nonvanishing characteristic functions such that the linear forms $L_{1}=\xi_{1}+\xi_{2}$ and $L_{2}=\xi_{1}+\alpha\xi_{2}$ are independent, i.e. the distribution of the random vector ${\boldsymbol{L}}=(L_{1},L_{2})$ belongs to the class $\mathcal{D}_{2,1}$. Condition $(\ref{12})$ in this case takes the form: $I-\alpha$ is a topological automorphism of the group $X$. This condition for such $\alpha$ is not true. As follows from the results of [12], the same is also true for the group $X=\mathbb{R}\times\mathbb{T}$.

In connection with the above, the following question arises. Let $X$ be an arbitrary second countable locally compact Abelian group, i.e. generally speaking, $X$ can contain a subgroup topologically isomorphic to the circle group $\mathbb{T}$. Assume that condition (2) or (12) in the case when coefficients $\alpha_{ji}$ are topological automorphisms of the group $X$, is valid. Does Theorem 1 hold true in this case? In other words, does Theorem 1 hold true if we omit the condition: the group $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$. It is interesting to remark that if $m=2$, i.e. we have two independent linear forms, the number of independent random variables $n=2$, and the coefficients $\alpha_{ji}$ are topological automorphisms of the group $X$, as it was proved in [9] the answer too this question is positive. **