Single-threshold maximal Baum–Katz characterizations for m-negatively associated random variables

Let $ S_n = \sum_{i = 1}^nX_i $ and $ M_n = \max_{1\le k\le n}|S_k| $. We establish a single-threshold converse for identically distributed $ m $-negatively associated ($ m $-NA) random variables. If, for some $ \varepsilon_0>0 $,

$ \sum\limits_{n\ge1}n^{q/p-2} \mathbb{P}(M_n>\varepsilon_0n^{1/p})<\infty, $

where $ p>0 $ and $ q\ge p $, then $ \mathbb{E}|X_1|^q < \infty $; moreover, for $ p>1 $, convergence at this single threshold also forces $ \mathbb{E} X_1 = 0 $. Consequently, when $ 1 < p < 2 $ and $ q\ge p $, the moment and centering conditions are equivalent both to maximal convergence at every positive threshold and to maximal convergence at some positive threshold. At the endpoint $ p = 1 $, we determine the threshold profile exactly: The infimum of the convergent thresholds equals $ | \mathbb{E} X_1| $. The proof combines a lower maximal-tail estimate on an interlaced NA subsequence with dyadic reconstruction and a weak-law drift-recovery argument, thereby separating tail integrability from deterministic drift. We also show that the maximum cannot be replaced by the terminal sum: A symmetric NA pair-cancellation construction has a summable terminal-sum series despite an infinite $ q $-th moment, whereas its maximal series diverges at least logarithmically. Finally, the relation with known sufficiency results is made explicit, and corresponding subcritical complete-moment and $ f $-moment consequences are recorded. Together these results give a maximal Baum–Katz characterization under the stated dependence structure.