Randomly Weighted Self-normalized Lévy Processes

الأصل · EN

Let (Uₜ,Vₜ) be a bivariate Lévy process, where Vₜ is a subordinator and Uₜ is a Lévy process formed by randomly weighting each jump of Vₜ by an independent random variable Xₜ having cdf F. We investigate the asymptotic distribution of the self-normalized Lévy process Uₜ/Vₜ at 0 and at ∞. We show that all subsequential limits of this ratio at 0 (∞) are continuous for any nondegenerate F with finite expectation if and only if Vₜ belongs to the centered Feller class at 0 (∞). We also characterize when Uₜ/Vₜ has a non-degenerate limit distribution at 0 and ∞.