##
**Functional equations and characterization problems on locally compact abelian groups.**
*(English)*
Zbl 1156.60003

EMS Tracts in Mathematics 5. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-045-6/hbk). xii, 256 p. (2008).

The emphasis of this monograph is laid on ‘classical’ characterizations of Gaussian laws on the real line resp. on finite dimensional vector spaces and the possibility of extensions to locally compact abelian groups. In fact, there exist various characterizations of Gaussian laws, the characterizing properties are all equivalent in the ‘classical’ setup. But even for the torus many of these properties are known to be not equivalent. Therefore, on general abelian groups, we have various definitions of Gaussian laws. Hence natural questions arise: E.g., describe the class of locally compact groups on which the properties under consideration are equivalent with a given definition of Gaussian laws.

This monograph is devoted to three closely related characterizations. The first, known as Kac-Bernstein-characterization: Let \(\xi_i\) be independent real random variables. Their distribution is Gaussian iff \(\left(\xi_1+\xi_2, \xi_1-\xi_2\right)\) are independent. Or, in the language of their characteristic functions \(f_i\), this is the case iff the following functional equation is satisfied: \[ \forall\;u,v\in \mathbb{R},\;f_1(u+v)f_2(u-v) = f_1(u)f_1(v)f_2(u)f_2(-v) \] A closely related characterization is known as theorem of Skitovich-Damois: Let \(\xi_i, 1\leq i \leq n\) be independent real random variables and \(\alpha_i, \beta_j \in \mathbb{R}\backslash \left\{0\right\}\). Define the linear forms \(L_1 := \sum_i\alpha_i\xi_i\), \(L_2 := \sum_j\beta_j\xi_j\). Then the \(\xi_i\) are all Gaussian iff \(\left(L_1, L_2\right)\) are independent.

Equivalently, again translated to characteristic functions, iff \[ \forall u,v\in \mathbb{R},\;\quad \prod_jf_j\left(\alpha_ju+\beta_jv \right) = \prod_jf_j\left(\alpha_ju \right) \cdot \prod_jf_j\left(\beta_jv \right) \] The third characterization due to C.C. Heyde: Assume – with the notations introduced before – that \(\forall i\neq j\): \(\beta_i\alpha_i^{-1}- \beta_j\alpha_j^{-1} \neq 0\). Then Gaussian laws are characterized by the property that the conditional distribution of \(L_2\) given \(L_1\) is symmetric. In terms of characteristic functions, iff \[ \prod_jf_j\left(\alpha_ju+\beta_jv \right) = \prod_jf_j\left(\alpha_ju-\beta_jv \right) \] For vector spaces, generally, for locally compact abelian groups \(X\), the defining properties are formulated analogous, where \(\alpha_i, \beta_j\) are assumed to be automorphisms, and, in Heyde’s characterization, the crucial assumption is replaced by \(\beta_i\alpha_i^{-1}- \beta_j\alpha_j^{-1} \in \mathrm{Aut}(X)\).

As afore mentioned, there exist various other possible definitions of Gaussian laws, e.g., due to K. Urbanik, saying that a law is Gaussian iff for any character, the image is a Gaussian law on the torus, or, in the context of continuous convolution semigroups, a law is Gaussian iff the Lévy measure in the Lévy Khinchin representation vanishes. During the last 3 decades a series of investigations were published, by the author and others, clearifying the relations between these definitions and the afore mentioned characterizing properties of Kac-Bernstein, Skitovich-Damois and Heyde. The reader is also referred to the author’s related earlier monograph [Arithmetic of probability distributions and characterization problems on abelian groups. Translations of Mathematical Monographs. 116. (Providence), RI: American Mathematical Society (AMS). (1993; Zbl 0925.60012)]. In the monograph under review the author collects and re-arranges these previous investigations.

A survey of the content: Chapter I is concerned with basic probability theory and harmonic analysis on \(X\), presenting in particular typical examples of locally compact abelian groups.

Chapter II contains definitions and various properties of Gaussian laws on abelian groups.

Chapter III is concerned with the validity of the Kac-Bernstein property, in particular, with the characterization of all \(X\) such that the Kac-Bernstein propety implies that the underlying laws are Gaussian, or at least, \(K-\)invariant with Gaussian image on the quotient \(X/K\) (where \(K\) denotes a compact subgroup.)

Chapter IV and V are concerned with the Skitovich-Damois-property, where in IV it is assumed that the characteristic functions are without zeros. In V zeros are allowed, and a variety of different groups is investigated. In the latter case idempotent factors turn out to be important and – e.g. in the case of finite groups – new effects arise.

Chapter VI is concerned with Heyde’s property, again assuming first that characteristic functions are without zeros, and considering then the general case. In particular, also the case of discrete groups is investigated.

The book closes with an appendix considering general functional equations of the afore mentioned type, and, finally with a section containing comments and open problems.

This monograph is devoted to three closely related characterizations. The first, known as Kac-Bernstein-characterization: Let \(\xi_i\) be independent real random variables. Their distribution is Gaussian iff \(\left(\xi_1+\xi_2, \xi_1-\xi_2\right)\) are independent. Or, in the language of their characteristic functions \(f_i\), this is the case iff the following functional equation is satisfied: \[ \forall\;u,v\in \mathbb{R},\;f_1(u+v)f_2(u-v) = f_1(u)f_1(v)f_2(u)f_2(-v) \] A closely related characterization is known as theorem of Skitovich-Damois: Let \(\xi_i, 1\leq i \leq n\) be independent real random variables and \(\alpha_i, \beta_j \in \mathbb{R}\backslash \left\{0\right\}\). Define the linear forms \(L_1 := \sum_i\alpha_i\xi_i\), \(L_2 := \sum_j\beta_j\xi_j\). Then the \(\xi_i\) are all Gaussian iff \(\left(L_1, L_2\right)\) are independent.

Equivalently, again translated to characteristic functions, iff \[ \forall u,v\in \mathbb{R},\;\quad \prod_jf_j\left(\alpha_ju+\beta_jv \right) = \prod_jf_j\left(\alpha_ju \right) \cdot \prod_jf_j\left(\beta_jv \right) \] The third characterization due to C.C. Heyde: Assume – with the notations introduced before – that \(\forall i\neq j\): \(\beta_i\alpha_i^{-1}- \beta_j\alpha_j^{-1} \neq 0\). Then Gaussian laws are characterized by the property that the conditional distribution of \(L_2\) given \(L_1\) is symmetric. In terms of characteristic functions, iff \[ \prod_jf_j\left(\alpha_ju+\beta_jv \right) = \prod_jf_j\left(\alpha_ju-\beta_jv \right) \] For vector spaces, generally, for locally compact abelian groups \(X\), the defining properties are formulated analogous, where \(\alpha_i, \beta_j\) are assumed to be automorphisms, and, in Heyde’s characterization, the crucial assumption is replaced by \(\beta_i\alpha_i^{-1}- \beta_j\alpha_j^{-1} \in \mathrm{Aut}(X)\).

As afore mentioned, there exist various other possible definitions of Gaussian laws, e.g., due to K. Urbanik, saying that a law is Gaussian iff for any character, the image is a Gaussian law on the torus, or, in the context of continuous convolution semigroups, a law is Gaussian iff the Lévy measure in the Lévy Khinchin representation vanishes. During the last 3 decades a series of investigations were published, by the author and others, clearifying the relations between these definitions and the afore mentioned characterizing properties of Kac-Bernstein, Skitovich-Damois and Heyde. The reader is also referred to the author’s related earlier monograph [Arithmetic of probability distributions and characterization problems on abelian groups. Translations of Mathematical Monographs. 116. (Providence), RI: American Mathematical Society (AMS). (1993; Zbl 0925.60012)]. In the monograph under review the author collects and re-arranges these previous investigations.

A survey of the content: Chapter I is concerned with basic probability theory and harmonic analysis on \(X\), presenting in particular typical examples of locally compact abelian groups.

Chapter II contains definitions and various properties of Gaussian laws on abelian groups.

Chapter III is concerned with the validity of the Kac-Bernstein property, in particular, with the characterization of all \(X\) such that the Kac-Bernstein propety implies that the underlying laws are Gaussian, or at least, \(K-\)invariant with Gaussian image on the quotient \(X/K\) (where \(K\) denotes a compact subgroup.)

Chapter IV and V are concerned with the Skitovich-Damois-property, where in IV it is assumed that the characteristic functions are without zeros. In V zeros are allowed, and a variety of different groups is investigated. In the latter case idempotent factors turn out to be important and – e.g. in the case of finite groups – new effects arise.

Chapter VI is concerned with Heyde’s property, again assuming first that characteristic functions are without zeros, and considering then the general case. In particular, also the case of discrete groups is investigated.

The book closes with an appendix considering general functional equations of the afore mentioned type, and, finally with a section containing comments and open problems.

Reviewer: Wilfried Hazod (Dortmund)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

62E10 | Characterization and structure theory of statistical distributions |

43A05 | Measures on groups and semigroups, etc. |

43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |

43A35 | Positive definite functions on groups, semigroups, etc. |

39B52 | Functional equations for functions with more general domains and/or ranges |