ASYMPTOTIC ANALYSIS OF RANDOM WALKS - HEAVY-TAILED DISTRIBUTIONS

This book focuses on the asymptotic behaviour 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 usuallyprove 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. The first unified systematic exposition of the large deviations theory for heavy-tailed random walks. Computing the probabilities of these large deviations (rare events) is essential, in fields such as finance, networks and insurance; this books shows you how. Includes many never before published results on large deviations for random walks with non-identically distributed jumps.