We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander.
"First-passage times for random walks with nonidentically distributed increments." Ann. Probab. 46 (6) 3313 - 3350, November 2018. https://doi.org/10.1214/17-AOP1248