Limit distribution of the position of a semicontinuous process with negative infinite mean at the moment of exit from an interval (Q1118909)

Let \(\xi\) (t), \(t\geq 0\), be a homogeneous process with independent increments having negative jumps and infinite negative mean. The following statement is proved: If \(E\{\exp [s(\xi (t)-\xi (0))]\}=\exp \{tk(s)\}\) and \(k(s)=- s^{\alpha}L(1/s)\), \(s\to 0\), where \(0<\alpha <1\) and L is positive, slowly varying at infinity, then \[ \lim_{x\to \infty}P_ x\{\xi (\zeta)/x<z\}=(\sin \pi \alpha /\pi)\int^{1}_{0}(v^{\alpha -1}/(1- v-z)^{\alpha})dv, \] where \(P_ x\) is the probability under the condition \(\xi (0)=x\).