About time of reaching a high level by a random walk in a random environment

Let $(p_{i},q_{i}) $, $i\in \mathbb{Z}$, be a sequence of independent identically distributed pairs of random variables, where $p_{0}+q_{0}=1$ and, in addition, $p_{0}>0$ and $q_{0}>0 $ a.s. We consider a random walk in the random environment $(p_{i},q_{i}) $, $i\in \mathbb{Z}$. This means that in a fixed random environment a walking particle located at some moment $n$ at a state $i$ jumps at moment $n+1$ either to the state $(i+1)$ with probability $p_{i}$ or to the state $(i-1)$ with probability $q_{i}$. It is assumed that the distribution of the random variable $\log (q_{0}/p_{0})$ belongs (without centering) to the domain of attraction of the two-sided stable law with index $\alpha \in (0,2] $. Let $T_{n}$ be the first passage time of level $n$ by the mentioned random walk. We prove the invariance principle for the logarithm of the stochastic process $\{T_{\lfloor ns\rfloor},s\in [0,1] \}$ as $n\to \infty$. This result is based on the limit theorem for a branching process in a random environment which allows precisely one immigrant in each generation.