This book focuses on the asymptotic behavior of the probabilities of large deviations of the trajectories of random walks with 'heavy-tailed' (in particular, regularly varying, sub- and semiexponential) jump distributions. Large deviation probabilities are of great interest in numerous applied areas, typical examples being ruin probabilities in risk theory, error probabilities in mathematical statistics, and buffer-overflow probabilities in queueing theory. The classical large deviation theory, developed for distributions decaying exponentially fast (or even faster) at infinity, mostly uses analytical methods. If the fast decay condition fails, which is the case in many important applied problems, then direct probabilistic methods usually prove to be efficient. This monograph presents a unified and systematic exposition of the large deviation theory for heavy-tailed random walks. Most of the results presented in the book are appearing in a monograph for the first time. Many of them were obtained by the authors.
Introduction; 1. Preliminaries; 2. Random walks with jumps having no finite first moment; 3. Random walks with finite mean and infinite variance; 4. Random walks with jumps having finite variance; 5. Random walks with semiexponential jump distributions; 6. Random walks with exponentially decaying distributions; 7. Asymptotic properties of functions of distributions; 8. On the asymptotics of the first hitting times; 9. Large deviation theorems for sums of random vectors; 10. Large deviations in the space of trajectories; 11. Large deviations of sums of random variables of two types; 12. Non-identically distributed jumps with infinite second moments; 13. Non-identically distributed jumps with finite variances; 14. Random walks with dependent jumps; 15. Extension to processes with independent increments; 16. Extensions to generalised renewal processes; Bibliographic notes; Index of notations; Bibliography.