  The infimum of an integrated current is its extreme value against the
direction of its average flow. Using martingale theory, we show that the infima
of integrated edge currents in time-homogeneous Markov jump processes are
geometrically distributed, with a mean value determined by the effective
affinity measured by a marginal observer that only sees the integrated edge
current. In addition, we show that a marginal observer can estimate a finite
fraction of the average entropy production rate in the underlying
nonequilibrium process from the extreme value statistics in the integrated edge
current. The estimated average rate of dissipation obtained in this way equals
the above mentioned effective affinity times the average edge current.
Moreover, we show that estimates of dissipation based on extreme value
statistics can be significantly more accurate than those based on thermodynamic
uncertainty ratios, as well as those based on a naive estimator obtained by
neglecting nonMarkovian correlations in the Kullback-Leibler divergence of the
trajectories of the integrated edge current.
