Uniform Local Asymptotics for Lévy Processes with Subexponential Jumps

Abstract

This paper is devoted to unifying the uniform local large-deviation asymptotics for a centered Lévy process $X$ with subexponential jumps. Our results assert that for any $\theta ,{\delta }_0>0$ and $K\ge 0$, $$\underset{t\to \infty }{\lim }\underset{x\ge \theta t}{\sup }\underset{|y|\le Kb(x)}{\sup }\underset{\delta \in [{\delta }_0,\infty ]}{\sup }\underset{0<s\le t}{\sup }|\frac{\mathbf{P}(X_s\in (x-y,x-y+\delta ])}{s\cdot \mathbf{P}(X_1\in (x,x+\delta ])}-1|=0,$$ where the natural-scale function $b$ satisfies a polynomial growth condition. This provides a continuous-time and simultaneously uniform analogue of the results of Denisov et al. [Ann. Probab., 2008], while being established under a weaker moment assumption.

22 pages