Randomly weighted self-normalized Lévy processes

Let ( U t , V t ) be a bivariate Lévy process, where V t is a subordinator and U t is a Lévy process formed by randomly weighting each jump of V t by an independent random variable X t having cdf F . We investigate the asymptotic distribution of the self-normalized Lévy process U t / V t 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 V t belongs to the centered Feller class at 0 ( ∞ ). We also characterize when U t / V t has a non-degenerate limit distribution at 0 and ∞ .