In this paper we get some sufficient conditions for the finiteness or nonfiniteness of the passage-time moments for nonnegative discrete parameter processes. The developed criteria are closely connected with the well-known results of Foster for the ergodicity of Markov chains and are given in terms of sub(super)martingales. Then, as an application of the obtained results, we get explicit conditions for the finiteness or nonfiniteness of passage-time moments for reflected random walks in a quadrant with zero drift in the interior.
"Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant." Ann. Probab. 24 (2) 932 - 960, April 1996. https://doi.org/10.1214/aop/1039639371