Precision of stochastic currents bounded by entropy production and mean residual time

The thermodynamic uncertainty relation bounds the precision of a time-integrated current in a Markov jump process by the entropy production rate. The kinetic uncertainty relation bounds the same precision by the dynamical activity, which counts the jumps per unit time, and the clock uncertainty relation bounds it at least as tightly by the inverse of the mean residual time, which is the mean time from a random instant to the next jump. The thermodynamic-kinetic uncertainty relation combines entropy production with the activity, so, far from equilibrium, its bound exceeds that of the clock relation if the escape rates, which are the total rates of leaving each state, are unequal. We prove a bound on the long-time precision of every current in terms of the entropy production rate and the mean residual time. Out of equilibrium with unequal escape rates, the new bound is tighter than the thermodynamic, the clock and the thermodynamic-kinetic uncertainty relations. Inverted, the new bound gives a lower bound on the entropy production rate from a measured precision. On a cycle of states with unequal forward rates, the ratio of this lower bound to the entropy production rate tends to a positive limit as the backward rates shrink relative to the forward rates, whereas the same ratio tends to zero for the lower bounds from the thermodynamic and the thermodynamic-kinetic relations. For the net number of jumps across one edge, a published lower bound from recurrence times is tighter in most of the random networks that we test. The tightening of the new bound relative to the tightest of the thermodynamic, the clock and the thermodynamic-kinetic relations is small, with a median below ten percent at the parameter values that we test on cycles of ten states and on random networks of ten states. Computing the new lower bound requires the current's mean and variance and the times of all jumps, which give the mean residual time.