  We study Markov decision processes (MDPs) with a countably infinite number of
states. The $\limsup$ (resp. $\liminf$) threshold objective is to maximize the
probability that the $\limsup$ (resp. $\liminf$) of the infinite sequence of
directly seen rewards is non-negative. We establish the complete picture of the
strategy complexity of these objectives, i.e., the upper and lower bounds on
the memory required by $\varepsilon$-optimal (resp. optimal) strategies. We
then apply these results to solve two open problems from (Sudderth, Decisions
in Economics and Finance, 2020) about the strategy complexity of optimal
strategies for the expected $\limsup$ (resp. $\liminf$) payoff.
