  For many stochastic processes, the probability $S(t)$ of not-having reached a
target in unbounded space up to time $t$ follows a slow algebraic decay at long
times, $S(t)\sim S_0/t^\theta$. This is typically the case of symmetric compact
(i.e. recurrent) random walks. While the persistence exponent $\theta$ has been
studied at length, the prefactor $S_0$, which is quantitatively essential,
remains poorly characterized, especially for non-Markovian processes. Here we
derive explicit expressions for $S_0$ for a compact random walk in unbounded
space by establishing an analytic relation with the mean first-passage time of
the same random walk in a large confining volume. Our analytical results for
$S_0$ are in good agreement with numerical simulations, even for strongly
correlated processes such as Fractional Brownian Motion, and thus provide a
refined understanding of the statistics of longest first-passage events in
unbounded space.
