On Large Deviations of Multivariate Heavy-Tailed Random Walks

Let {Sn;n=1,2,…} be a random walk in Rd and E(S1)=(μ1,…,μd). Let aj>μj for j=1,…,d and A=(a1,∞)×⋅⋅⋅×(ad,∞). We are interested in the probability P(Sn/n∈A) for large n in the case where the components of S1 are heavy tailed. An objective is to associate an exact power with the aforementioned probability. We also derive sharper asymptotic bounds for the probability and show that in essence, the occurrence of the event {Sn/n∈A} is caused by large single increments of the components in a specific way.