Zaihraiev O. Large-deviation theorems for sums of independent and identically distributed random vectors. - Torun: University of Nicolaus Copernicus Press, 2005. - 187 p. - ISBN 83-223-1872-1  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Contents

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