In this paper, we consider two kinds of stochastic models with asymptotic perturbation. One is the non-nearest neighbor random walk with asymptotic perturbation, for which we consider the transience and recurrence of the random walk.We give the criterion for positive recurrent of the random walk. The other is the branching process in random environment with asymptotic perturbation,for which by analyzing the limit behavior of the associated random walk, we give the criterion of extinction and non-extinction of the branching process.

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.