Asymptotic results for heavy-tailed Lévy processes and their exponential functionals

In this paper, we first provide several conditional limit theorems for Lévy processes with negative drift and regularly varying tail. Then apply them to study the asymptotic behavior of expectations some exponential functionals heavy-tailed processes. As key point, observe that mainly depends on sample paths early arrival large jump. Both polynomial decay rate exact expression coefficients are given. an application, give description extinction speed continuous-state branching in random environment stable mechanism.