Preface to the second enlarged edition .......................... v
Preface ....................................................... vii
1 Probability Measures on Metric Spaces ........................ 1
1.1 Tight measures .......................................... 1
1.2 The topology of weak convergence ........................ 5
1.3 The Prokhorov theorem .................................. 18
1.4 Convolution of measures ................................ 23
2 The Fourier Transform in a Banach Space ..................... 29
2.1 Fourier transforms of probability measures ............. 29
2.2 Shift compact sets of probability measures ............. 39
2.3 Infinitely divisible and embeddable measures ........... 50
2.4 Gauss and Poisson measures ............................. 56
3 The Structure of Infinitely Divisible Probability
Measures .................................................... 71
3.1 The Ito-Nisio theorem .................................. 71
3.2 Fourier expansion and construction of Brownian
motion ................................................. 86
3.3 Symmetric Lévy measures and generalized Poisson
measures ............................................... 98
3.4 The Lévy-Khinchin decomposition ....................... 114
4 Harmonic Analysis of Convolution Semigroups ................ 133
4.1 Convolution of Radon measures ......................... 133
4.2 Duality of locally compact Abelian groups ............. 144
4.3 Positive definite functions ........................... 162
4.4 Positive definite measures ............................ 171
5 Negative Definite Functions and Convolution Semigroups ..... 185
5.1 Negative definite functions ........................... 185
5.2 Convolution semigroups and resolvents ................. 191
5.3 Lévy functions ........................................ 204
5.4 The Lévy-Khinchin representation ...................... 211
6 Probabilistic Properties of Convolution Semigroups ......... 225
6.1 Transient convolution semigroups ...................... 225
6.2 The transience criterion .............................. 237
6.3 Recurrent random walks ................................ 253
6.4 Classification of transient random walks .............. 272
7 Hypergroups in Probability Theory .......................... 291
7.1 Commutative hypergroups ............................... 291
7.2 Decomposition of convolution semigroups of measures ... 303
7.3 Random walks in hypergroups ........................... 318
7.4 Increment processes and convolution semigroups ........ 333
8 Limit Theorems on Locally Compact Abelian Groups ........... 345
8.1 Limit problems and parametrization of weakly
infinitely divisible measures ......................... 345
8.2 Gaiser's limit theorem ................................ 349
8.3 Limit theorems for symmetric arrays and Bernoulli
arrays ................................................ 360
8.4 Limit theorems for special locally compact Abelian
groups ................................................ 366
Appendices .................................................... 375
A Topological groups ......................................... 375
В Topological vector spaces .................................. 377
С Commutative Banach algebras ................................ 383
Selected References ........................................... 389
Symbols ....................................................... 397
Index ......................................................... 403
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