Large deviations for transient random walks with asymptotically zero drifts

We consider transient nearest-neighbor random walks $X:=\{X_n{\}}_{n\ge 0}$ 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 $M_t:=max\{X_i,0\le i\le t\}$ be the maximum process of $X$, and $T_n:=inf\{t\ge 0,X_t=n\}$ 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.