Large deviations for random walks under subexponentiality: the big-jump domain

Stimulated by applications to internet traffic modeling and insurance mathematics, distributions with heavy tails have been widely studied over the past decades. This talk addresses a fundamental large-deviation problem for random walks with heavy-tailed step-size distributions. We consider so-called subexponential step-size distributions, which constitute the most widely-used class of heavy-tailed distributions. I will present a theory to study sequences {x_n} for which P{S_n>x_n} behaves asymptotically like n P {S_1>x_n} for large n. (joint work with D. Denisov and V. Shneer)