The Uniform Local Asymptyotics of the Overshoot of a Random Walk with Heavy-Tailed Increments

It is well known that much attention has been paid to issues of the uniform local asymptotics of the overshoot of a random walk with heavy-tailed increments. Asmussen et al. (2003) has got a special result of local asymptotics of the overshoot of a random walk which has some conditions of moment. Rencently, Tang (2007) has investigated the non-local asymptotics of the overshoot of a random walk. In the first chapter of this paper, we introduce some fundmential knowledge about our paper. Furthermore, we fall back a lot of results which have been got before and give the aim of our paper. In the second chapter, we obtain the uniform local asymptotics of the overshoot of a random walk with heavy-tailed increments. When the overshoot is in (x + y, x + y + T) for some 0＜T＜∞, we discuss the asymptotics for two cases about y, respectively. Firstly, y≥f(x) for any positive function f(x)→∞as x→∞; secondly, y∈[0, N] for any 0＜N＜∞. Based on above results, we discuss them further and get a result of the uniform local asymptotics of the overshoot of a random walk for the case y≥0. The obtained results require certain uniform requirements for y. At last of this paper, we give a corollary and some remarks. We illustrate that the two results of the second chapter are meaningful by the corollary, and these remarks show the direction of our further research.