Characterization theorems for Q-independent random variables in Banach spaces

TL;DR

This paper explores the properties and characterization theorems of Q-independent random variables within Banach spaces, aiming to deepen understanding of their structure and behavior in functional analysis.

Contribution

It introduces new characterization theorems for Q-independent random variables in Banach spaces, expanding the theoretical framework of their analysis.

Findings

Established new characterization theorems for Q-independence in Banach spaces

Enhanced understanding of the structure of Q-independent random variables

Provided foundational results for further research in functional analysis

Abstract

Characterization theorems for Q-independent random variables in Banach spaces

Equations

$$\hat{\mu }(f)=\mathbf{E}[e^{i<\xi ,f>}]=\underset{X}{\int }{e}^{i<x,f>}d\mu (x),f\in X^{\ast }.$$

$$\hat{\mu }(f)=e^{i<m,f>-\frac{1}{2}<Rf,f>},f\in X^{\ast },$$

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

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

$${\hat{\mu }}_{({\xi }_1,\dots ,{\xi }_n)}(f_1,\dots ,f_n)=\mathbf{E}[({\xi }_1,f_1)\cdots ({\xi }_n,f_n)]=$$

$$=\left(\prod _{j=1}^n{\hat{\mu }}_{{\xi }_j}(f_j)\right)\exp \{q(f_1,\dots ,f_n)\},f_j\in X^{\ast },$$

$$\prod _{j=1}^n{\hat{\mu }}_j(A_j^{\ast }f+B_j^{\ast }g)=\prod _{j=1}^n{\hat{\mu }}_j(A_j^{\ast }f)\prod _{j=1}^n{\hat{\mu }}_j(B_j^{\ast }g)\exp \{r(f,g)\},f,g\in X^{\ast },$$

$$\hat{\mu }(f)=e^{\psi (f)},$$

$$\prod _{j=1}^n{\hat{\mu }}_j(f+C_j^{\ast }g)=\prod _{j=1}^n{\hat{\mu }}_j(f)\prod _{j=1}^n{\hat{\mu }}_j(C_j^{\ast }g)\exp \{r(f,g)\},f,g\in X^{\ast },$$

$$\sum _{j=1}^n{\psi }_j(f+C_j^{\ast }g)=P(f)+Q(g)+r(f,g),f,g\in X^{\ast },$$

$$P(f)=\sum _{j=1}^n{\psi }_j(f),Q(g)=\sum _{j=1}^n{\psi }_j(C_j^{\ast }g).$$

$$\sum _{j=1}^{n-1}{\Delta }_{l_{n,j}}{\psi }_j(f+C_j^{\ast }g)={\Delta }_{h_n}P(f)+{\Delta }_{k_n}Q(g)+{\Delta }_{(h_n,k_n)}r(f,g),f,g\in X^{\ast },$$

$$\sum _{j=1}^{n-2}{\Delta }_{l_{n-1,j}}{\Delta }_{l_{n,j}}{\psi }_j(f+C_j^{\ast }g)={\Delta }_{h_{n-1}}{\Delta }_{h_n}P(f)+{\Delta }_{k_{n-1}}{\Delta }_{k_n}Q(g)+{\Delta }_{(h_{n-1},k_{n-1})}{\Delta }_{(h_n,k_n)}r(f,g),f,g\in X^{\ast },$$

$${\Delta }_{l_{2,1}}{\Delta }_{l_{3,1}}\dots {\Delta }_{l_{n,1}}{\psi }_1(f+C_1^{\ast }g)=$$

$$={\Delta }_{h_2}{\Delta }_{h_3}\dots {\Delta }_{h_n}P(f)+{\Delta }_{k_2}{\Delta }_{k_3}\dots {\Delta }_{k_n}Q(g)+{\Delta }_{(h_2,k_2)}{\Delta }_{(h_3,k_3)}\dots {\Delta }_{(h_n,k_n)}r(f,g),f,g\in X^{\ast },$$

$${\Delta }_{h_1}{\Delta }_{h_2}\dots {\Delta }_{h_n}P(f)+{\Delta }_{k_1}{\Delta }_{k_2}\dots {\Delta }_{k_n}Q(g)+{\Delta }_{(h_1,k_1)}{\Delta }_{(h_2,k_2)}\dots {\Delta }_{(h_n,k_n)}r(f,g)=0,f,g\in X^{\ast }.$$

$${\Delta }_h{\Delta }_{h_1}{\Delta }_{h_2}\dots {\Delta }_{h_n}P(f)+{\Delta }_{(h,0)}{\Delta }_{(h_1,k_1)}{\Delta }_{(h_2,k_2)}\dots {\Delta }_{(h_n,k_n)}r(f,g)=0,f,g\in X^{\ast }.$$

$${\Delta }_{(h,k)}^{l+1}r(f,g)=0,f,g\in X^{\ast },$$

$${\Delta }_h^{l+2}{\Delta }_{h_1}{\Delta }_{h_2}\dots {\Delta }_{h_n}P(f)=0,f\in X^{\ast }.$$

$${\Delta }_h^{n+l+2}P(f)=0,f,h\in X^{\ast }.$$

$$\prod _{j=1}^n{\hat{\mu }}_j(A_j^{\ast }f+B_j^{\ast }g)=\prod _{j=1}^n{\hat{\mu }}_j(A_j^{\ast }f-B_j^{\ast }g)\exp \{r(f,g)\},f,g\in X^{\ast },$$

$$\prod _{j=1}^n{\hat{\mu }}_j(f+C_j^{\ast }g)=\prod _{j=1}^n{\hat{\mu }}_j(f-C_j^{\ast }g)\exp \{r(f,g)\},f,g\in X^{\ast },$$

$$\sum _{j=1}^{4n}{\lambda }_j(V)\subset U,$$

$$\sum _{j=1}^n{\psi }_j(f+C_j^{\ast }g)=\sum _{j=1}^n{\psi }_j(f-C_j^{\ast }g)+r(f,g),f,g\in V,$$

$$\sum _{j=1}^n{\Delta }_{l_{n,j}}{\psi }_j(f+C_j^{\ast }g)=\sum _{j=1}^{n-1}{\Delta }_{m_{n,j}}{\psi }_j(f-C_j^{\ast }g)+{\Delta }_{(h_n,h_n)}r(f,g),f,g\in V,$$

$$\sum _{j=1}^n{\Delta }_{l_{n-1,j}}{\Delta }_{l_{n,j}}{\psi }_j(f+C_j^{\ast }g)=\sum _{j=1}^{n-2}{\Delta }_{m_{n-1,j}}{\Delta }_{m_{n,j}}{\psi }_j(f-C_j^{\ast }g)+{\Delta }_{(h_{n-1},h_{n-1})}{\Delta }_{(h_n,h_n)}r(f,g),f,g\in V,$$

$$\sum _{j=1}^n{\Delta }_{l_{1,j}}{\Delta }_{l_{2,j}}\dots {\Delta }_{l_{n,j}}{\psi }_j(f+C_j^{\ast }g)={\Delta }_{(h_1,h_1)}{\Delta }_{(h_2,h_2)}\dots {\Delta }_{(h_n,h_n)}r(f,g),f,g\in V,$$

$$\sum _{j=1}^{n-1}{\Delta }_{b_{n,j}}{\Delta }_{l_{1,j}}{\Delta }_{l_{2,j}}\dots {\Delta }_{l_{n,j}}{\psi }_j(f+C_j^{\ast }g)={\Delta }_{(C_n^{\ast }{h}_n,-h_n)}{\Delta }_{(h_1,h_1)}{\Delta }_{(h_2,h_2)}\dots {\Delta }_{(h_n,h_n)}r(f,g),f,g\in V,$$

$$\sum _{j=1}^{n-2}{\Delta }_{b_{n-1,j}}{\Delta }_{b_{n,j}}{\Delta }_{l_{1,j}}{\Delta }_{l_{2,j}}\dots {\Delta }_{l_{n,j}}{\psi }_j(f+C_j^{\ast }g)$$

$$={\Delta }_{(C_{n-1}^{\ast }{k}_{n-1},-k_{n-1})}{\Delta }_{(C_n^{\ast }{h}_n,-h_n)}{\Delta }_{(h_1,h_1)}{\Delta }_{(h_2,h_2)}\dots {\Delta }_{(h_n,h_n)}r(f,g),f,g\in V,$$

Full text

M.V.Myronyuk

B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine

myronyuk@ilt.kharkov.ua

CHARACTERIZATION THEOREMS FOR Q-INDEPENDENT RANDOM VARIABLES IN BANACH SPACE

Let $X$ be a Banach space. Following A. Kagan and G. Szekely we introduce a notion of $Q$-independence for random variables with values in $X$. We prove analogues of the Skitovich–Darmois and Heyde theorems for $Q$-independent random variables with values in X.

1. Introduction. In 1953 V.P.Skitovich and G.Garmois proved independently that a Gaussian distribution on a real line is characterized by the independence of two linear forms from $n$ independent random variables ([1, $\S$3.1]). In 1962 S.G.Ghurye and I.Olkin generalized the Skitovich-Darmois theorem for random variables with values in $\mathbb{R}^{n}$ and nondegenerate matrices as the coefficients of forms ([1, $\S$3.2]).

The Skitovich-Darmois theorem and other characterization theorems were generalized in different algebraical structures, particularly, on locally compact Abelian groups and Banach spaces (see e.g. [2]-[6]).

In 1985 W. Krakowiak generalized the Skitovich–Darmois theorem for random variables with values in a separable Banach space $X$ and linear continuous invertible operators as the coefficients of forms ([7]). In the paper [8] this result was proved in another way. Besides the Heyde theorem for Banach spaces was proved in [8].

In the paper [9] A. Kagan and G. Székely introduced a notion of $Q$-independence of random variables which generalizes a notion of independence and proved that some classical theorems hold true if instead of independence $Q$-independence is considered. Particularly they proved that the Skitovich-Darmois theorem and the Cramer theorem about decomposition of a Gaussian distribution hold true for $Q$-independent random variables. In the paper [10] A.Il’inskii studied polynomials which can appear in a notion of $Q$-independence. In the paper [11] B.L.S. Prakasa Rao proved a generalization of the Kotlarski theorem (see [12]) for $Q$-independent random variables with values in a Hilbert space. Some generalizations of the Skitovich-Darmois theorem and the Heyde theorem for $Q$-independent random variables with values in locally compact Abelian groups were proved in [13] and [14].

A notion of $Q$-independence can be introduced in a natural way in a Banach space. We prove that the Skitovich-Darmois theorem and the Heyde theorem in a Banach space hold true if we consider $Q$-independence instead of independence (Theorems 1 and 2). These results generalize the main results of the paper [8]. Proofs in these cases follow the scheme of analogous theorems of the paper [8]. Also we prove another characterization theorems (Theorems 3 and 4) for $Q$-independent random variables with values in a Banach space. In contrast to Theorems 1 and 2, Theorem 3 and 4 are new not only for $Q$-independent random variables, but also for independent random variables.

2. Notation and definitions. We use standard facts of the theory of probability distributions in a Banach space (see, e.g., [15]). Let $X$ be a separable real Banach space and let $X^{*}$ be the space dual to it. Denote by ${\rm GL}(X)$ the set all linear continuous invertible operators in $X$. Denote by $A^{*}$ the adjoint operator to a linear continuous operator $A$. Denote by $I$ the identity operator. Denote by $<~{}x,f>$ the value of a functional $f\in X^{*}$ at an element $x\in X$. Denote by $\mathcal{M}^{1}(X)$ the convolution semigroup of probability distributions on $X$. For $\mu\in\mathcal{M}^{1}(X)$ we denote by $\overline{\mu}$ the distribution defined by the formula $\overline{\mu}(E)=\mu(-E)$ for any Borel set $E\subset X$. The characteristic functional of the probability distribution $\mu$ of a random variable $\xi$ with values in $X$ is defined by the formula

[TABLE]

Note that $\widehat{\overline{\mu}}(f)={\overline{\widehat{\mu}}}(f)$. It is known that $\widehat{\mu}(f)$ is positive definite and sequentially continuous in the pointwise topology ([15, Ch.4, §2]).

In the paper, unless otherwise specified, we work in a strong topology, i.e. the topology arising from a norm.

Definition 1. A random variable $\xi$ is called Gaussian if either it is degenerated or for any $f\in X^{*}$ the random variable $<\xi,f>$ is Gaussian.

The following proposition follows immediately from Definition 1.

Proposition 1. A distribution $\mu$ on $X$ is Gaussian iff for any $f\in X^{*}$ the characteristic function $\widehat{\mu}_{f}(t)$ of a random variable $<\xi,f>$ is a characteristic function of a Gaussian distribution on $\mathbb{R}$.

Note that the following proposition is true in a Banach space.

Proposition 2 ([15, Ch. 4, $\S$2]). Let $X$ be a real Banach space. The characteristic functional of a Gaussin distribution $\mu\in\mathcal{M}^{1}(X)$ has the form

[TABLE]

where $m\in X$, and $R:X^{*}\longrightarrow X$ is a symmetric nonnegative operator. The element $m$ is called the mean value of the distribution $\mu$, and $R$ is called its covariation operator.

Conversely, if $\mu\in\mathcal{M}^{1}(X)$ has a characteristic functional of the form $(\ref{0.2})$, where $m\in X$ is a certain element and $R:X^{*}\longrightarrow X$ is a certain symmetric nonnegative operator, then $\mu$ is a Gaussian distribution in $X$ with mean value $m$ and covariation operator $R$.

Let $\psi(f)$ be a function on $X^{*}$, and let $h$ be an arbitrary element of $X^{*}$. We denote by $\Delta_{h}$ an operator of finite difference

[TABLE]

A function $\psi(f)$ on $X^{*}$ is called a polynomial if

[TABLE]

for some $n$ and for all $f,h\in X^{*}$.

Let $\xi_{1},\dots,\xi_{n}$ be random variables with values in separable Banach space $X$. Following A. Kagan and G. Székely ([9]), we say that random variables $\xi_{1},\dots,\xi_{n}$ are $Q$-independent if the characteristic functional of the distribution of the vector $(\xi_{1},\dots,\xi_{n})$ can be represented in the form

[TABLE]

[TABLE]

where $q(f_{1},\dots,f_{n})$ is a continuous polynomial on $(X^{*})^{n}$ such that $q(0,\dots,0)=0$.

$Q$-analogue of the Cramer theorem in a Banach space follows directly from Proposition 1 and the $Q$-analogue of the Cramer theorem on a real line (see [9]).

Proposition 3. If a random variable $\xi$ with values in a real separable Banach space $X$ has a Gaussian distribution and $\xi=\xi_{1}+\xi_{2}$, where $\xi_{1},\xi_{2}$ are $Q$-independent random variables, then $\xi_{1},\xi_{2}$ have Gaussian distributions too.

3. The Skitovich-Darmois theorem. We prove an analogue of the Skitovich-Darmois theorem for $Q$-independent random variables in a Banach space.

Theorem 1. Let $\xi_{1},\dots,\xi_{n},\ n\geq 2,$ be $Q$-independent random variables with values in a real separable Banach space $X$ and with distributions $\mu_{j}$. Let $A_{j},B_{j}\in GL(X)$. If linear forms $L_{1}=A_{1}\xi_{1}+\dots+A_{n}\xi_{n}$ and $L_{2}=B_{1}\xi_{1}+\dots+B_{n}\xi_{n}$ are $Q$-independent then $\mu_{j}$ are Gaussian distributions.

To prove Theorem 1 we need the following lemmas.

Lemma 1. Let $\xi_{1},\dots,\xi_{n},\ n\geq 2,$ be $Q$-independent random variables with values in a real separable Banach space $X$ and with distributions $\mu_{j}$. Let $A_{j},B_{j}$ be linear continuous operators in $X$. Linear forms $L_{1}=A_{1}\xi_{1}+\dots+A_{n}\xi_{n}$ and $L_{2}=B_{1}\xi_{1}+\dots+B_{n}\xi_{n}$ are $Q$-independent iff the characteristic functionals $\widehat{\mu}_{j}(y)$ satisfy the equation

[TABLE]

where $r(f,g)$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0)=0$.

The proof of Lemma 1 repeats literally the proof of the similar lemma for locally compact Abelian groups of the paper [13].

Lemma 2 ([8]). If $\mu$ is a Gaussian distribution in a real separable Banach space $X$ and $\mu=\mu_{1}*\mu_{2}$, then $\mu_{1}$ and $\mu_{2}$ are Gaussian distributions on $X$ too.

Lemma 3 ([8]). Let $X$ be a real separable Banach space, $\mu\in\mathcal{M}^{1}(X)$. If

[TABLE]

where $\psi(f)$ is a polynomial, in a certain neighborhood of zero, then $\mu$ is a Gaussian distribution on $X$.

Proof of Theorem 1. Putting $\xi^{\prime}_{j}=A_{j}\xi_{j}$, we reduce the proof to the case of linear forms $L_{1}=\xi_{1}+\dots+\xi_{n}$ and $L_{2}=C_{1}\xi_{1}+\dots+C_{n}\xi_{n}$, where $C_{j}\in{\rm GL}(X)$. It follows from Lemma 1 that the condition of $Q$-independence of $L_{1}$ and $L_{2}$ is equivalent to the statement that the characteristic functionals $\widehat{\mu}_{j}(y)$ satisfy the equation

[TABLE]

where $r(f,g)$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0)=0$. Thus, the proof of Theorem 1 reduces to the description of solutions of equation (5) in the class of characteristic functionals of probability distributions on $X^{*}$.

Set $\nu_{j}=\mu_{j}*\bar{\mu}_{j}$. Then $\widehat{\nu}_{j}(f)=|\widehat{\mu}_{j}(f)|^{2}\geq 0$. It is obvious that the characteristic functionals $\widehat{\nu}_{j}(f)$ satisfy equation (5) too. If we prove that $\widehat{\nu}_{j}(f)$ are characteristic functionals of Gaussian distributions, then Lemma 2 implies that $\widehat{\mu}_{j}(f)$ are characteristic functionals of Gaussian distributions too. Therefore we can assume from the beginning that $\widehat{\mu}_{j}(f)\geq 0$.

Since all $\widehat{\mu}_{j}(f)\geq 0$, we have $\exp\{r(f,g)\}>0$. The verification of the fact that $\widehat{\mu}_{j}(f)>0$ for all $f\in X^{*}$, $j=1,2,\dots,n$, is the same as in the paper [8].

We show that $\widehat{\mu}_{j}(f)$ is a characteristic functional of a Gaussian distribution. Set $\psi_{j}(f)=-\log\widehat{\mu}_{j}(f)$. It follows from (5) that

[TABLE]

where

[TABLE]

We use the finite differences method to solve (6) and we follow the scheme of the proof of the Skitovich-Darmois theorem for locally compact Abelian groups (see [17]). Let $h_{n}$ be an arbitrary element of $X^{*}$. We set $k_{n}=-(C_{n}^{*})^{-1}h_{n}$. Then $h_{n}+C_{n}^{*}k_{n}=0$. Substitute in (6) $f+h_{n}$ for $f$ and $g+k_{n}$ for $g$. Subtracting equation (6) from the resulting equation we obtain

[TABLE]

where $l_{n,j}=h_{n}+C^{*}_{j}k_{n}=(C^{*}_{j}-C^{*}_{n})k_{n}$, $j=1,2,\dots,n-1$. We note that the left-hand side of (7) does not contain the function $\psi_{n}$. Let $h_{n-1}$ be an arbitrary element of $X^{*}$. We set $k_{n-1}=-(C^{*}_{n-1})^{-1}h_{n-1}$. Then $h_{n-1}+C^{*}_{n-1}k_{n-1}=0$. Substitute in (6) $f+h_{n-1}$ for $f$ and $g+k_{n-1}$ for $g$. Subtracting equation (7) from the resulting equation we obtain

[TABLE]

where $l_{n-1,j}=h_{n-1}+C^{*}_{j}k_{n-1}=(C^{*}_{j}-C^{*}_{n-1})k_{n-1}$, $j=1,2,\dots,n-2$. The left-hand side of (8) does not contain functions $\psi_{n}$ and $\psi_{n-1}$. Reasoning similarly, we get the equation

[TABLE]

[TABLE]

where $h_{m}$ is an arbitrary element of $X^{*}$, $k_{m}=-(C^{*}_{m})^{-1}h_{m}$, $l_{m,1}=h_{m}+C^{*}_{1}k_{m}=(C^{*}_{1}-C^{*}_{m})k_{m}$, $m=2,3,\dots,n$. Let $h_{1}$ be an arbitrary element of $X^{*}$. We set $k_{1}=-(C^{*}_{1})^{-1}h_{1}$. Then $h_{1}+C^{*}_{1}k_{1}=0$. Substitute in (S0.Ex6) $f+h_{1}$ for $f$ and $g+k_{1}$ for $g$. Subtracting equation (S0.Ex6) from the resulting equation we obtain

[TABLE]

Let $h$ be an arbitrary element of $X^{*}$. Substitute in (10) $f+h$ for $f$. Subtracting equation (10) from the resulting equation we obtain

[TABLE]

Since $r(f,g)$ is a polynomial, we have

[TABLE]

for some $l$ and arbitrary $h$ and $k$ from $X^{*}$.

We apply $\Delta^{l+1}_{(h,k)}$ to (11). Taking into account (12), we get

[TABLE]

Note that $f,h$ and $h_{m}$, $m=1,..,n$, are arbitrary elements of $X^{*}$. We can put in (11) $h_{1}=\dots=h_{n}=h$. Then

[TABLE]

Thus, $P(f)$ is a polynomial on $X^{*}$. We denote $\gamma=\mu_{1}*\dots*\mu_{n}$. Then $\widehat{\gamma}(f)=\prod_{j=1}^{n}\widehat{\mu}_{j}(f)$. Thus, $\widehat{\gamma}(f)=e^{-P(f)}$. It follows from Lemma 3 that $\widehat{\gamma}(f)$ is a characteristic functional of a Gaussian distribution. Then it follows from Proposition 3 that $\widehat{\mu}_{j}(f)$ is a characteristic functional of a Gaussian distribution. Theorem 1 is proved. $\blacksquare$

4. The Heyde theorem. We prove an analogue of the Heyde theorem for $Q$-independent random variables in a Banach space.

Theorem 2. Let $\xi_{1},\dots,\xi_{n},\ n\geq 2,$ be $Q$-independent random variables with values in a real separable Banach space $X$ and with distributions $\mu_{j}$. Let $A_{j},B_{j}\in{\rm GL}(X)$ such that $B_{i}A_{i}^{-1}\pm B_{j}A_{j}^{-1}\in{\rm GL}(X)$ for all $i\neq j$. If the conditional distribution of the linear form $L_{2}=B_{1}\xi_{1}+\dots+B_{n}\xi_{n}$ given $L_{1}=A_{1}\xi_{1}+\dots+A_{n}\xi_{n}$ is symmetric, then $\mu_{j}$ are Gaussian distributions.

To prove Theorem 2 we need the following lemma.

Lemma 4. Let $\xi_{1},\dots,\xi_{n},\ n\geq 2,$ be $Q$-independent random variables with values in a real separable Banach space $X$ and with distributions $\mu_{j}$. Let $A_{j},B_{j}\in{\rm GL}(X)$. The conditional distribution of the linear form $L_{2}=B_{1}\xi_{1}+\dots+B_{n}\xi_{n}$ given $L_{1}=A_{1}\xi_{1}+\dots+A_{n}\xi_{n}$ is symmetric iff the characteristic functional $\widehat{\mu}_{j}(y)$ satisfy equation

[TABLE]

where $r(f,g)$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0)=0$.

The proof of Lemma 4 repeats literally the proof of the similar lemma for locally compact Abelian groups of the paper [13].

Proof of Theorem 2. As in the proof of Theorem 1, we show that the proof of Theorem 2 reduces to the case of $L_{1}=\xi_{1}+\dots+\xi_{n}$ and $L_{2}=C_{1}\xi_{1}+\dots+C_{n}\xi_{n}$, where $C_{j}\in{\rm GL}(X)$ such that $C_{i}\pm C_{j}\in{\rm GL}(X)$ for all $i\neq j$, and characteristic functional $\widehat{\mu}_{j}(f)\geq 0$. It follows from Lemma 4 that the condition of symmetry of the conditional distribution of $L_{2}$ given $L_{1}$ is equivalent to the statement that the characteristic functionals $\widehat{\mu}_{j}(y)$ satisfy the equation

[TABLE]

where $r(f,g)$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0)=0$.

Since $\widehat{\mu}_{j}(0)=1$ and $\widehat{\mu}_{j}(f)$ are sequentially continuous, there exists a neighborhood $U$ of zero such that $\widehat{\mu}_{j}(y)>0$ for all $y\in U$. We choose in $U$ a symmetric neighborhood $V$ of zero such that

[TABLE]

where $\lambda_{j}\in\{I,C^{*}_{1},\dots,C^{*}_{n}\}$.

We take the logarithm of equation (16) in the neighborhood $V$ and get

[TABLE]

where $\psi_{j}(f)=-\log\widehat{\mu}_{j}(f)$. We use the finite differences method to solve (17) and we follow the scheme of the proof of the classical Heyde (see [1, $\S$13.4.1]).

Let $h_{n}$ be an arbitrary element of $V$. Substitute in (17) $f+C_{n}^{*}h_{n}$ for $f$ and $g+h_{n}$ for $g$. Subtracting equation (17) from the resulting equation we obtain

[TABLE]

where $l_{n,j}=(C_{n}^{*}+C^{*}_{j})h_{n}$, $m_{n,j}=(C_{n}^{*}-C^{*}_{j})h_{n}$. Note that the right-hand side of (18) does not contain the function $\psi_{n}$. Let $h_{n-1}$ be an arbitrary element of $V$. Substitute in (18) $f+C^{*}_{n-1}h_{n-1}$ for $f$ and $g+h_{n-1}$ for $g$. Subtracting equation (18) from the resulting equation we obtain

[TABLE]

where $l_{n-1,j}=(C_{n-1}^{*}+C^{*}_{j})h_{n-1}$, $m_{n,j}=(C_{n-1}^{*}-C^{*}_{j})h_{n-1}$. The right-hand side of (19) does not contain the functions $\psi_{n}$ and $\psi_{n-1}$. Reasoning similarly, we get the equation

[TABLE]

where $h_{i}$ is an arbitrary element of $V$, $l_{i,j}=(C_{i}^{*}+C^{*}_{j})h_{i}$, $i=1,2,\dots,n$.

Let $k_{n}$ be an arbitrary element of $V$. Substitute in (20) $f+C_{n}^{*}k_{n}$ for $f$ and $g-k_{n}$ for $g$. Subtracting equation (20) from the resulting equation we obtain

[TABLE]

where $b_{n,j}=(C_{n}^{*}-C^{*}_{j})k_{n}$. Note that the left-hand side of (21) does not contain the function $\psi_{n}$. Let $k_{n-1}$ be an arbitrary element of $V$. Substitute in (21) $f+C^{*}_{n-1}k_{n-1}$ for $f$ and $g-k_{n-1}$ for $g$. Subtracting equation (21) from the resulting equation we obtain

[TABLE]

[TABLE]

where $b_{n-1,j}=(C_{n-1}^{*}-C^{*}_{j})k_{n-1}$. The left-hand side of (S0.Ex8) does not contain the functions $\psi_{n}$ and $\psi_{n-1}$. Reasoning similarly, we get the equation

[TABLE]

[TABLE]

where $k_{i}$ is an arbitrary element of $V$, $b_{i,1}=(C_{i}^{*}-C^{*}_{1})k_{i}$, $i=2,3,\dots,n$.

Since $r(f,g)$ is a polynomial, the equality (12) holds true for some $l$ and arbitrary $h$ and $k$ from $X^{*}$. Taking it into account, we apply the operator $\Delta^{l+1}_{(h,k)}$ to the both sides of (S0.Ex9). Putting $g=0$ in the obtained equation, we get

[TABLE]

Since $k_{i},h_{i}$ are arbitrary elements of $V$, $C_{i}^{*}\pm C_{j}^{*}\in{\rm GL}(X^{*})$ for all $i\neq j$, and $l_{i,j}=(C_{i}^{*}+C^{*}_{j})h_{i}$, $b_{i,1}=(C_{i}^{*}-C^{*}_{1})k_{i}$, we get in some neighborhood $W$ of zero that

[TABLE]

We obtain that $\widehat{\mu}_{1}(f)=e^{-\psi_{1}(f)}$, where $\psi_{1}(f)$ is a polynomial, in the neighborhood $W$ of zero. It follows from Lemma 3 that $\mu_{1}$ is a Gaussian distribution. Similarly we obtain that all $\mu_{j}$ are Gaussian distributions. $\blacksquare$

If in Theorem 2 $n=2$, then the condition on coefficients of linear forms can be relaxed. Indeed, it is shown in the proof of Theorem 2 that we can assume that for $n=2$ the conditional distribution of $L_{2}=C_{1}\xi_{1}+C_{2}\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric. Apply the operator $C_{1}^{-1}$ to $L_{2}$. It is obvious that the conditional distribution of $C_{1}^{-1}L_{2}$ given $L_{1}$ is symmetric too. Thus we can assume that $L_{1}=\xi_{1}+\xi_{2}$ and $L_{2}=\xi_{1}+C\xi_{2}$.

The following statement holds true.

Theorem 3. Let $\xi_{1},\xi_{2}$ be $Q$-independent random variables with values in a real separable Banach space $X$ and with distributions $\mu_{j}$. Let $C\in{\rm GL}(X)$ such that $C+I\in{\rm GL}(X)$. If the conditional distribution of the linear form $L_{2}=\xi_{1}+C\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then $\mu_{j}$ are Gaussian distributions.

To prove Theorem 3 we need the following lemma.

Lemma 5. Let $\xi_{1},\xi_{2}$ be $Q$-independent random variables with values in a real separable Banach space $X$ and with distributions $\mu_{j}$. Let $C_{j}\in{\rm GL}(X)$. If the conditional distribution of the linear form $L_{2}=C_{1}\xi_{1}+C_{2}\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then $L^{\prime}_{1}=(C_{1}+C_{2})\xi_{1}+2C_{2}\xi_{2}$ and $L^{\prime}_{2}=2C_{1}\xi_{1}+(C_{1}+C_{2})\xi_{2}$ are $Q$-independent.

Proof. By Lemma 4 the characteristic functionals $\widehat{\mu}_{j}(y)$ satisfy the equation

[TABLE]

where $r(f,g)$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0)=0$.

Putting $f=C_{2}^{*}h,g=-h$ and then $f=-C_{1}^{*}h,g=h$ in equation (26), we get

[TABLE]

[TABLE]

Let $k,l\in X^{*}$. Put $f=C_{1}^{*}l+C_{2}^{*}k$, $g=k+l$ in (26). We obtain

[TABLE]

[TABLE]

Taking into account (27) and (28), we can rewrite equation (S0.Ex10) in the form

[TABLE]

[TABLE]

[TABLE]

It follows from Lemma 1 and equation (S0.Ex11) that the linear forms $L^{\prime}_{1}=(C_{1}+C_{2})\xi_{1}+C_{2}\xi_{2}$ and $L^{\prime}_{2}=2C_{1}\xi_{1}+(C_{1}+C_{2})\xi_{2}$ are $Q$-independent. $\blacksquare$

Proof of Theorem 3. It follows from Lemma 5 that the linear forms $L^{\prime}_{1}=(I+C)\xi_{1}+2C\xi_{2}$ and $L^{\prime}_{2}=2\xi_{1}+(I+C)\xi_{2}$ are $Q$-independent. Since $I+C\in{\rm GL}(X)$, all coefficients of linear forms $L^{\prime}_{1}$ and $L^{\prime}_{2}$ are linear continuous invertible operators. It follows from Theorem 1 that Theorem 3 holds true. $\Box$

Remark 1. Suppose that in Theorem 3 a Banach space $X$ is reflexive. It is not difficult to verify that the condition $I+C\in{\rm GL}(X)$ can be relaxed to the condition $Ker(I+C)=\{0\}$, and the assertion of the theorem does not change. Indeed, in this case $\overline{Im(I+C^{*})}=X^{*}$. It follows from the proof of Theorem 1 that only this condition is essential.

Since the independence of random variables implies their $Q$-independence, it follows from Theorem 3 the following statement.

Corollary 1. Let $\xi_{1},\xi_{2}$ be independent random variables with values in a real separable Banach space $X$ and with distributions $\mu_{j}$. Let $C\in{\rm GL}(X)$ such that $C+I\in{\rm GL}(X)$. If the conditional distribution of the linear form $L_{2}=\xi_{1}+C\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then $\mu_{j}$ are Gaussian distributions.

5. The independence of the sample mean and the residue vector. It is well known that the Gaussian distribution on a real line is characterized by the independence of the sample mean and the residue vector. Sufficiency of this statement is obvious. Necessity follows from the Geary-Lucacs-Laga theorem ([1, $\S$4.2]). This characterization of a Gaussian distribution was generalized on locally compact Abelian groups in the paper [16]. In this section we prove an analogue of this characterization for $Q$-independent random variables with values in a Banach space.

Theorem 4. Let $\xi_{1},\dots,\xi_{n},\ n\geq 2,$ be $Q$-independent identically distributed random variables with values in a real separable Banach space $X$ and with a distribution $\mu$. Put ${S={1\over n}\sum_{j=1}^{n}\xi_{j}}$ and ${\bf V}=(\xi_{1}-S,\dots,\xi_{n}-S)$. If ${S}$ and ${\bf V}$ are $Q$-independent then $\mu$ is a Gaussian distribution.

To prove Theorem 4 we need the following lemma.

Lemma 6. Let $\xi_{1},\dots,\xi_{n},\ n\geq 2,$ be $Q$-independent identically distributed random variables with values in a real separable Banach space $X$ and with a distribution $\mu$. Put ${S={1\over n}\sum_{j=1}^{n}\xi_{j}}$ and ${\bf V}=(\xi_{1}-S,\dots,\xi_{n}-S)$. If ${S}$ and ${\bf V}$ are $Q$-independent then the characteristic functional $\widehat{\mu}(f)$ satisfy equation

[TABLE]

[TABLE]

where $r(f,g_{1},\dots,g_{n})$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0,\dots,0)=0$.

Proof. Note that $S$ and ${\bf V}$ are $Q$-independent iff the equality

[TABLE]

holds for all $u,v_{1},\dots,v_{n}\in X^{*}$, where $q_{1}(f,g_{1},\dots,g_{n})$ is a continuous polynomial on $(X^{*})^{2}$, $q_{1}(0,0,\dots,0)=0$. Since random variables $\xi_{1},\dots,\xi_{n}$ are $Q$-independent, the left-hand side of (32) can be represented in the form

[TABLE]

[TABLE]

[TABLE]

Analogously we transform the right-hand side of (32):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Put

[TABLE]

Lemma is proved. $\Box$

Proof of Theorem 4. It follows from Lemma 6 that the function $\widehat{\mu}(f)$ satisfies equation (S0.Ex13). Substituting $f=nh,g_{1}=g,\ g_{2}=-g,g_{3}=\dots=g_{n}=0$ in (S0.Ex13), we obtain

[TABLE]

where $r(f,g_{1},\dots,g_{n})$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0,\dots,0)=0$.

Let $n=2$. Equation (33) takes the form of equation (5) for $n=2$ and $C_{1}=I$, $C_{2}=-I$. Since it follows from Lemma 1 that the linear forms $L_{1}=\xi_{1}+\xi_{2}$ and $L_{2}=\xi_{1}-\xi_{2}$ are independent, Theorem 4 follows in this case from the proof of Theorem 1.

Let $n\geq 3$. We can write equation (33) in the form

[TABLE]

where $\widetilde{r}(f,g)$ is a continuous polynomial on $(X^{*})^{2}$, $r(0,0)=0$.

It follows from (34) that the set

[TABLE]

is an open subgroup of $X^{*}$. Thus $B=X^{*}$ and $\widehat{\mu}(f)\neq 0$ on $X^{*}$. As in the proof of Theorem 1, we can show that we can assume from the beginning that $\widehat{\mu}_{j}(f)>0$. Then we also have that $r(f,g_{1},\dots,g_{n})>0$.

Put $\psi(f)=-\log\widehat{\mu}(f)$. Taking the logarithm of (S0.Ex13) and multiplying all arguments of functions in (S0.Ex13) on $n$, we get

[TABLE]

[TABLE]

We use the finite difference method to solve equation (S0.Ex24). Let $h_{1}$ be an arbitrary element of $X^{*}$. Substitute $f+h_{1}$ for $f$ and $g_{j}+h_{1}$ for $g_{j}$, $j=1,2,\dots,n$ in equation (S0.Ex24). Subtracting equation (S0.Ex24) from the resulting equation we obtain

[TABLE]

Нехай $h_{2}$ — це довiльний елемент $X^{*}$. Покладемо $f+h_{2}$ замiсть $f$ та $g_{1}+h_{2}$ замiсть $g_{1}$ в рiвняннi (S0.Ex24). Вiднiмаючи (36) з отриманого рiвняння, ми маємо

[TABLE]

[TABLE]

Let $h_{3}$ be an arbitrary element of $X^{*}$. Substitute $f+h_{3}$ for $f$ and $g_{2}+h_{3}$ for $g_{2}$ in equation (S0.Ex24). Subtracting equation (S0.Ex25) from the resulting equation we obtain

[TABLE]

[TABLE]

Since $r(f,g_{1},\dots,g_{n})$ is a polynomial, we have

[TABLE]

for some $l$ and arbitrary elements $k$ and $k_{j}$ from $X^{*}$.

Applying the operator $\Delta^{l+1}_{(k,k_{1},\dots,k_{n})}$ to (S0.Ex26) and taking into account (39), we get

[TABLE]

Put $h_{1}=h_{2}=h_{3}=k=h$ in (40). We obtain

[TABLE]

i.e. $\psi(y)$ is a continuous polynomial. Thus, $\widehat{\mu}(f)=e^{-\psi(f)}$. It follows from Lemma 3 that $\widehat{\mu}(f)$ is a characteristic functional of a Gaussian distribution. Theorem 4 is proved. $\blacksquare$

Since the independence of random variables implies their $Q$-independence, it follows from Theorem 4 the following statement.

Corollary 2. Let $\xi_{1},\dots,\xi_{n},\ n\geq 2,$ be independent identically distributed random variables with values in a real separable Banach space $X$ and with a distribution $\mu$. Put ${S={1\over n}\sum_{j=1}^{n}\xi_{j}}$ and ${\bf V}=(\xi_{1}-S,\dots,\xi_{n}-S)$. If ${S}$ and ${\bf V}$ are independent then $\mu$ is a Gaussian distribution.