  We consider a one-dimensional stationary time series of fixed duration $T$.
We investigate the time $t_{\rm m}$ at which the process reaches the global
maximum within the time interval $[0,T]$. By using a path-decomposition
technique, we compute the probability density function $P(t_{\rm m}|T)$ of
$t_{\rm m}$ for several processes, that are either at equilibrium (such as the
Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with
stochastic resetting). We show that for equilibrium processes the distribution
of $P(t_{\rm m}|T)$ is always symmetric around the midpoint $t_{\rm m}=T/2$, as
a consequence of the time-reversal symmetry. This property can be used to
detect nonequilibrium fluctuations in stationary time series. Moreover, for a
diffusive particle in a confining potential, we show that the scaled
distribution $P(t_{\rm m}|T)$ becomes universal, i.e., independent of the
details of the potential, at late times. This distribution $P(t_{\rm m}|T)$
becomes uniform in the "bulk" $1\ll t_{\rm m}\ll T$ and has a nontrivial
universal shape in the "edge regimes" $t_{\rm m}\to0$ and $t_{\rm m} \to T$.
Some of these results have been announced in a recent Letter [Europhys. Lett.
{\bf 135}, 30003 (2021)].
