  We study the statistics of first passage times (FPTs) of trajectory
observables in both classical and quantum Markov processes. We consider
specifically the FPTs of counting observables, that is, the times to reach a
certain threshold of a trajectory quantity which takes values in the positive
integers and is non-decreasing in time. For classical continuous-time Markov
chains we rigorously prove: (i) a large deviation principle (LDP) for FPTs,
whose corollary is a strong law of large numbers; (ii) a concentration
inequality for the FPT of the dynamical activity, which provides an upper bound
to the probability of its fluctuations to all orders; and (iii) an upper bound
to the probability of the tails for the FPT of an arbitrary counting
observable. For quantum Markov processes we rigorously prove: (iv) the quantum
version of the LDP, and subsequent strong law of large numbers, for the FPTs of
generic counts of quantum jumps; (v) a concentration bound for the the FPT of
total number of quantum jumps, which provides an upper bound to the probability
of its fluctuations to all orders, together with a similar bound for the
sub-class of quantum reset processes which requires less strict irreducibility
conditions; and (vi) a tail bound for the FPT of arbitrary counts. Our results
allow to extend to FPTs the so-called "inverse thermodynamic uncertainty
relations" that upper bound the size of fluctuations in time-integrated
quantities. We illustrate our results with simple examples.
