Uniform Local Asymptotics for L\'evy Processes with Subexponential Jumps

Abstract

This paper is devoted to unifying the uniform local large-deviation asymptotics for a centered L\'evy process $X$ with subexponential jumps. Our results assert that for any $\theta,\delta_0>0$ and $K\geq0$, $$\lim_{t\to\infty}\sup_{x\geq\theta t}\sup_{|y|\leq Kb(x)}\sup_{\delta\in[\delta_0,\infty]}\sup_{0<s\leq t}\bigg|\frac{\mathbf P\big(X_s\in(x-y,x-y+\delta]\big)}{s\cdot\mathbf P\big(X_1\in(x,x+\delta]\big)}-1\bigg|=0,$$ 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.