Functional Equations and Characterization Problems on Locally Compact Abelian Groups

Preface......Page 6
Contents......Page 12
1 Locally compact Abelian groups......Page 14
2 Probability distributions on locally compact Abelian groups......Page 24
3 Properties of Gaussian distributions......Page 32
4 Cramér's theorem on the decomposition of a Gaussian distribution......Page 45
5 Polynomials on locally compact Abelian groups and the Marcinkiewicz theorem......Page 51
6 Gaussian distributions in the sense of Urbanik......Page 62
7 Locally compact Abelian groups for which the Kac–Bernstein theorem holds......Page 69
8 Random variables with values in the group R x T and in the a-adic solenoid Sigma_a......Page 82
9 Gaussian distributions in the sense of Bernstein......Page 94
10 Locally compact Abelian groups for which the Skitovich–Darmois theorem holds......Page 105
11 Random variables with values in the two-dimensional torus T^2......Page 120
12 Random variables with values in the groups R x T and _Sigma_a x T......Page 134
13 The number of random variables n=2......Page 146
14 The number of random variables n > 3......Page 166
15 Random variables with values in the a-adic solenoid Sigma_a......Page 185
16 The characteristic functions of random variables do not vanish......Page 199
17 Random variables with values in finite and discrete Abelian groups......Page 210
Appendix. The Kac–Bernstein and Skitovich–Darmois functional equations on locally compact Abelian groups......Page 236
Comments and unsolved problems......Page 250
Bibliography......Page 260
Symbol index......Page 266
Subject index......Page 268