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 $θ,δ_0>0$ and $K\geq0$, $$\lim_{t\to\infty}\sup_{x\geqθt}\sup_{|y|\leq Kb(x)}\sup_{δ\in[δ_0,\infty]}\sup_{0<s\leq t}\bigg|\frac{\mathbf P\big(X_s\in(x-y,x-y+δ]\big)}{s\cdot\mathbf P\big(X_1\in(x,x+δ]\big)}-1\bigg|=0,$$ 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.