Herein boundary crossing behavior of conditional random walks is studied. Asymptotic distributions of the exit time and the excess over the boundary are derived. In the course of derivation, two results of independent interest are also obtained: Lemma 4.1 shows that a conditional random walk behaves like an unconditional one locally in a very strong sense. Theorem B.1 describes a class of distributions over which the renewal theorem holds uniformly. Applications are given for modified repeated significance tests and change-point problems.
Inchi Hu. "Nonlinear Renewal Theory for Conditional Random Walks." Ann. Probab. 19 (1) 401 - 422, January, 1991. https://doi.org/10.1214/aop/1176990553