Given a real-valued random variable $X$ whose Laplace transform is
analytic in a neighbourhood of 0, we consider a random walk
${(S_{n},n\geq 0)}$, starting from the origin and with increments
distributed as $X$. We investigate the class of stopping times $T$
which are independent of $S_{T}$ and standard, i.e. $(S_{n\wedge
T},n\geq 0)$ is uniformly integrable. The underlying filtration
$(\mathcal{F}_{n},n\geq 0)$ is not supposed to be natural. Our
research has been deeply inspired by \cite{De Meyer-Roynette-Vallois-Yor 2002},
where the analogous problem is studied, but not yet solved, for the Brownian
motion. Likewise, the classification of all possible distributions
for $S_{T}$ remains an open problem in the discrete setting, even
though we manage to identify the solutions in the special case
where $T$ is a stopping time in the natural filtration of a
Bernoulli random walk and $\min T \le 5$. Some examples illustrate
our general theorems, in particular the first time where $|S_{n}|$
(resp. the age of the walk or Pitman's process) reaches a given
level $a\in\mathbb{N}^{\ast}$. Finally, we are concerned with a
related problem in two dimensions. Namely, given two independent
random walks $(S_{n}^{\prime},n\geq 0)$ and $(S_{n}^{\prime\prime},n\geq 0)$
with the same incremental distribution, we search for stopping times
$T$ such that $S_{T}^{\prime}$ and $S_{T}^{\prime\prime}$ are independent.