Introduction .................................................... 5
Chapter 1. Large-deviation theorems for heavy-tailed
distributions attracted by normal law ............... 25
1.1. Regularly varying absolutely continuous distributions ..... 25
1.1.1. Theorems and corollaries ........................... 27
1.1.2. Proofs ............................................. 31
1.2. Regularly varying lattice distributions ................... 48
1.2.1. Theorems and corollaries ........................... 48
1.2.2. Proofs ............................................. 50
Chapter 2. Multivariate a-stable distributions ................. 61
2.1. Analytical properties of α-stable distributions ........... 61
2.1.1. Asymptotic formulas for α-stable densities ......... 62
2.1.2. Proofs ............................................. 71
2.2. Large-deviation theorems for regularly varying
absolutely continuous distributions attracted
by α-stable law ........................................... 80
2.2.1. Theorems and corollaries ........................... 83
2.2.2. Proofs ............................................. 86
2.3. Large-deviation theorems for singular directions .......... 97
2.3.1. Theorems and corollaries ........................... 98
2.3.2. Auxiliary statements .............................. 100
2.3.3. Proofs ............................................ 105
Chapter 3. Limit theorems under the Cramer condition .......... 119
3.1. Gamma-like distributions ................................. 119
3.1.1. Abel-type theorem ................................. 121
3.1.2. Limit laws for conjugate distributions ............ 123
3.1.3. Local limit theorems .............................. 125
3.1.4. Proofs ............................................ 128
3.2. Compactly supported distributions ........................ 142
3.2.1. Abel-type theorem ................................. 142
3.2.2. Limit laws for conjugate distributions ............ 144
3.2.3. Local limit theorems .............................. 146
3.2.4. Remarks and example ............................... 148
3.2.5. Proofs ............................................ 154
Appendix A. Regularly varying functions and regularly
varying distributions ............................. 165
Appendix B. Auxiliary result .................................. 172
Bibliography .................................................. 177
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