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].