Joint distribution of the first hitting time and first hitting place for a random walk

Abstract

A random walk on the real line starting from 0 is considered. A representation of the Lapalace-Fourier transform of the joint distribution of the first hitting time and the first hitting place of the set (−∞, −a) (a>0) is obtained, which gives a relation with the joint distribution of those of the set (−∞, 0). The leading idea is Wiener-Hopf's factorization theorem.