  Recent large deviation results have provided general lower bounds for the
fluctuations of time-integrated currents in the steady state of stochastic
systems. A corollary are so-called thermodynamic uncertainty relations
connecting precision of estimation to average dissipation. Here we consider
this problem but for counting observables, i.e., trajectory observables which,
in contrast to currents, are non-negative and non-decreasing in time (and
possibly symmetric under time reversal). In the steady state, their
fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson
distribution dependent only on the averages of the observable and of the
dynamical activity. We show how to obtain the corresponding bounds for
first-passage times (times when a certain value of the counting variable is
first reached) and their uncertainty relations. Just like entropy production
does for currents, dynamical activity controls the bounds on fluctuations of
counting observables.
