Randomly Weighted Self-normalized L\'evy Processes

Let $(U_t,V_t)$ be a bivariate L'evy process, where $V_t$ is a subordinator and $U_t$ is a L'evy 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 L'evy process $U_t\mathrm{/}{V}_t$ at 0 and at $\mathrm{\infty }$. We show that all subsequential limits of this ratio at 0 ($\mathrm{\infty }$) are continuous for any nondegenerate $F$ with finite expectation if and only if $V_t$ belongs to the centered Feller class at 0 ($\mathrm{\infty }$). We also characterize when $U_t\mathrm{/}{V}_t$ has a non-degenerate limit distribution at 0 and $\mathrm{\infty }$.