Simple symmetric random walk in tight strip

A simple symmetric random walk $S_0:=0,\ S_n:= S_{n-1}+X_n,\ n\in\mathbb{N}$ is considered. Here $X_1,X_2,\ldots$ are independent and identically distributed random variables taking values $1,\ -1$ with probability $0.5$. Denote $ A_n(N):=\{0 \le S_i\le N,\ i=1,2,\ldots, n\}. $ The further conditions supposed to be true: $ N(n)\in \mathbb{N},\ N(n)\to\infty,\ N(n) = o\left(\sqrt{n}\right),\ n\to\infty. $
The next results will be presented in the talk: precise asymptotics of the probability ${\mathbf P}(A_n(N(n)))$ is obtained and the conditional limit theorem on the convergence of finite-dimensional distributions of the process $\left\{S_{[tn]},\ t\in[0,1]\right\}$ given $A_n(N(n))$.