  We consider an active run-and-tumble particle (RTP) in $d$ dimensions and
compute exactly the probability $S(t)$ that the $x$-component of the position
of the RTP does not change sign up to time $t$. When the tumblings occur at a
constant rate, we show that $S(t)$ is independent of $d$ for any finite time
$t$ (and not just for large $t$), as a consequence of the celebrated Sparre
Andersen theorem for discrete-time random walks in one dimension. Moreover, we
show that this universal result holds for a much wider class of RTP models in
which the speed $v$ of the particle after each tumbling is random, drawn from
an arbitrary probability distribution. We further demonstrate, as a
consequence, the universality of the record statistics in the RTP problem.
