  One of the characteristic features of a stochastic process under resetting is
that the probability density converges to a nonequilibrium stationary state
(NESS). In addition, the approach to the stationary state exhibits a dynamical
phase transition, which can be interpreted as a traveling front separating
spatial regions for which the probability density has relaxed to the NESS from
those where it has not. One can establish the existence of the phase transition
by carrying out an asymptotic expansion of the exact solution. In this paper we
develop an alternative, direct method for characterizing the approach to the
NESS of a stochastic process with resetting that is based on the calculation of
the so-called accumulation time. The latter is the analog of the mean first
passage time of a search process, in which the survival probability density is
replaced by an accumulation fraction density. In the case of one-dimensional
Brownian motion with Poissonian resetting, we derive the asymptotic formula
$|x-x_0|\approx \sqrt{4rD}T(x)$ for $|x-x_0|\gg\sqrt{D/r}$, where $T(x)$ is the
accumulation time at $x$, $r$ is the constant resetting rate, $D$ is the
diffusivity and $x_0$ is the reset point. This is identical in form to the
traveling front condition for the dynamical phase transition. We also derive an
analogous result for diffusion in higher spatial dimensions and for
non-Poissonian resetting. We then consider the effects of delays such as
refractory periods and finite return times. In both cases we establish that the
asymptotic behavior of $T(x)$ is independent of the delays. Finally, we extend
the analysis to a run-and-tumble particle with resetting. We thus establish the
accumulation time of a stochastic process with resetting as a useful quantity
for characterizing the approach to an NESS (if it exists) that is relatively
straightforward to calculate.
