Functional Equations and Characterization Problems on Locally Compact Abelian Groups

Preface......Page 5
Contents......Page 11
1 Locally compact Abelian groups......Page 13
2 Probability distributions on locally compact Abelian groups......Page 23
3 Properties of Gaussian distributions......Page 31
4 Cramér's theorem on the decomposition of a Gaussian distribution......Page 44
5 Polynomials on locally compact Abelian groups and the Marcinkiewicz theorem......Page 50
6 Gaussian distributions in the sense of Urbanik......Page 61
7 Locally compact Abelian groups for which the Kac–Bernstein theorem holds......Page 68
8 Random variables with values in the group R x T and in the a-adic solenoid Sigma_a......Page 81
9 Gaussian distributions in the sense of Bernstein......Page 93
10 Locally compact Abelian groups for which the Skitovich–Darmois theorem holds......Page 104
11 Random variables with values in the two-dimensional torus T^2......Page 119
12 Random variables with values in the groups R x T and _Sigma_a x T......Page 133
13 The number of random variables n=2......Page 145
14 The number of random variables n > 3......Page 165
15 Random variables with values in the a-adic solenoid Sigma_a......Page 184
16 The characteristic functions of random variables do not vanish......Page 198
17 Random variables with values in finite and discrete Abelian groups......Page 209
Appendix. The Kac–Bernstein and Skitovich–Darmois functional equations on locally compact Abelian groups......Page 235
Comments and unsolved problems......Page 249
Bibliography......Page 259
Symbol index......Page 265
Subject index......Page 267