|Abstract:||The acquisition of phonotactics goes through an early developmental stage of pure phonotactics learning, throughout which the learner is blind to alternations, as morphology lags behind. Furthermore, the acquisition of phonotactics is gradual, as the target adult phonotactics is slowly approached through a path of more conservative intermediate stages. This gradualness suggests an online learning model: the learner entertains over time a current hypothesis of the target phonotactics; this hypothesis is initialized to the most unmarked grammar; and it is slightly updated each time it is found to be incompatible with the current piece of data. This paper and the companion paper Magri (2010a) investigate online models of the early stage of the acquisition of phonotactics within the framework of Optimality Theory (OT), from a computational perspective. This paper focuses on convergence. It argues that demotion-only OT online algorithms are unsuited to the task of modeling the early stage: because of lack of alternations, the learner posits faithful underlying forms; the faithfulness constraints thus never make any mistake; and they are therefore never re-ranked by demotion-only; the resulting ranking dynamics is thus too simple to model acquisitional data. The paper thus tackles the problem of devising provably convergent re-ranking rules for OT online algorithms that perform constraint promotion too. This problem is solved, showing that convergence can be achieved by keeping the promotion amounts small. The solution rests on a previously unnoticed connection between the notion of OT-compatibility and the geometric notion of conic independence. The companion paper Magri (2010a) then turns to correctness. It shows that the problem of the acquisition of phonotactics cannot be solved in its full generality, namely not without restrictive assumptions on the underlying OT typologies. The paper thus develops tools to study correctness of OT online algorithms based on properties of the constraint set and the candidate sets. In particular, it shows that even a small amount of constraint promotion suffices to ensure correctness in simple but nontrivial cases.