Large deviations for transient random walks with asymptotically zero drifts

We consider transient nearest-neighbor random walks $ X: = \{X_{n}\}_{n\geq0} $ on the half-line, whose transition probabilities are state-dependent with certain asymptotic perturbations. This is a specific case of Lamperti's random walks. Let $ M_{t}: = \max\{X_{i}, \; 0\leq i\leq t\} $ be the maximum process of $ X $, and $ T_{n}: = \inf\{t\geq0, X_{t} = n\} $ be its inverse process. Hong&Yang (2019) provided the law of large numbers for $ T_{n} $. In this paper, we study the large deviations for $ M_{t} $ and $ T_{n} $ with speeds less than $ n $. This indicates that the perturbations slow down the random walk.