Let $(X_{n}) _{n\in \Bbb{N}}$ be a sequence of i.i.d.\ random variables with common density $f(x)$ such that the cumulant generating function $\log \Bbb{E}(e^{tX_{1}})$ is finite on some interval $[ 0,t_{0})$ which guarantees that all the moments of $X_1$ are finite. Consider a random walk $S_{n}=X_{1}+\cdots +X_{n}$, $n\geq 1$. The paper focuses on the asymptotics of the conditional distribution of $(X_{1},\ldots ,X_{n}) $ or $( X_{1},\ldots ,X_{k})$, as $n\rightarrow \infty$, conditioned on the event $\{ S_{n}\geq nb\}$ (or $\{ S_{n}=nb\}$) that the random walk exceeds (or is at) a level $bn$ with $b>\Bbb{E}X_{1}$.