Large deviation probabilities for random walks with heavy tails (Q1876391)

The authors consider the sums and maxima of sequential sums of random variables with regularly varying distribution tails of index \(-\beta\), with \(\beta\in (1,2)\). First-order approximations are established as well as some refinements for the distributions of these sums and maxima in a large deviation area. Similar results are obtained for probabilities of crossing arbitrary curvilinear boundaries in the large deviation range by random walks generated by such sequential sums.