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Recent large deviation results have provided general lower bounds for thefluctuations of time-integrated currents in the steady state of stochasticsystems. A corollary are so-called thermodynamic uncertainty relationsconnecting precision of estimation to average dissipation. Here we considerthis problem but for counting observables, i.e., trajectory observables which,in contrast to currents, are non-negative and non-decreasing in time (andpossibly symmetric under time reversal). In the steady state, theirfluctuations to all orders are bound from below by a Conway-Maxwell-Poissondistribution dependent only on the averages of the observable and of thedynamical activity. We show how to obtain the corresponding bounds forfirst-passage times (times when a certain value of the counting variable isfirst reached) and their uncertainty relations. Just like entropy productiondoes for currents, dynamical activity controls the bounds on fluctuations ofcounting observables.

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables