  We consider a model of space-continuous one-dimensional random walk with
simple correlation between the steps: the probability that two consecutive
steps have same sign is $q$ with $0\leq q\leq 1$. The parameter $q$ allows thus
to control the persistence of the random walk. We compute analytically the
survival probability of a walk of $n$ steps, showing that it is independent of
the jump distribution for any finite $n$. This universality is a consequence of
the Sparre-Andersen theorem for random walks with uncorrelated and symmetric
steps. We then apply this result to derive the distribution of the step at
which the random walk reaches its maximum and the record statistics of the
walk, which show the same universality. In particular, we show that the
distribution of the number of records for a walk of $n\gg 1$ steps is the same
as for a random walk with $n_{\rm eff}(q)=n/(2(1-q))$ uncorrelated and
symmetrically distributed steps. We also show that in the regime where $n\to
\infty$ and $q\to 1$ with $y=n(1-q)$, this model converges to the
run-and-tumble particle, a persistent random walk often used to model the
motion of bacteria. Our theoretical results are confirmed by numerical
simulations.
