Randomly Weighted Self-normalized L\'evy Processes

TL;DR

This paper studies the asymptotic behavior of a self-normalized bivariate Lévy process where one component is a subordinator and the other is a weighted sum of its jumps, revealing conditions for continuous limits and non-degenerate distributions.

Contribution

It characterizes the conditions under which the ratio of the weighted Lévy process to the subordinator converges to continuous or non-degenerate limits at 0 and infinity.

Findings

All subsequential limits are continuous under certain conditions.

Characterization of when the ratio has a non-degenerate limit distribution.

Conditions for the process to belong to the centered Feller class.

Abstract

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 $U_t/V_t$ at 0 and at $\infty$. We show that all subsequential limits of this ratio at 0 ($\infty$) are continuous for any nondegenerate $F$ with finite expectation if and only if $V_t$ belongs to the centered Feller class at 0 ($\infty$). We also characterize when $U_t/V_t$ has a non-degenerate limit distribution at 0 and $\infty$.