General criteria of integrability of functions of passage-times for non-negative stochastic processes and their applications

In this paper we study the question of integrability of functions of the first passage-times into compact sets and first return-times for stochastic processes with discrete parameter. We consider first a class of processes with negative drifts taking values in $\mathbb{R}_{+}$ and prove for them general sufficient conditions for integrability of functions of these random times. The conditions are formulated in a martingale spirit initiated by Foster and generalize corresponding results obtained earlier. In the second part of the paper we address a similar question for reflected random walks in a quadrant with zero-drift in the interior. Applying the results of the first part we get conditions for integrability of certain functions of the first passage-times and the first return-times for the reflected random walks. The obtained estimates provide quite sharp results for the former random times and complement the corresponding results in [S. Aspandiiarov and R. Iasnogorodski, Tails of passage-time for non-negative stochastic processes and an application to stochastic processes with boundary reflection in a wedge, Stochastic Process. Appl., 66 (1997), pp. 115–145]. Finally, we derive bounds for the rate of convergence of transition probabilities of ergodic reflected random walks to the corresponding invariant measure.