Let (Sn)n≥0 be a $\mathbb Z$-random walk and
$(\xi_{x})_{x\in \mathbb Z}$ be a sequence of independent and
identically distributed $\mathbb R$-valued random variables,
independent of the random walk. Let h be a measurable, symmetric
function defined on $\mathbb R^2$ with values in $\mathbb R$. We study the
weak convergence of the sequence ${\cal U}_{n}, n\in \mathbb N$, with
values in D[0,1] the set of right continuous real-valued
functions
with left limits, defined by
\[
\sum_{i,j=0}^{[nt]}h(\xi_{S_{i}},\xi_{S_{j}}), t\in[0,1].
\]
Statistical applications are presented, in particular we prove a strong law of large numbers
for U-statistics indexed by a one-dimensional random walk using a result of [1].