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This paper provides a Central Limit Theorem (CLT) for a process $\{\theta_n,n\geq 0\}$ satisfying a stochastic approximation (SA) equation of the form$\theta_{n+1} = \theta_n + \gamma_{n+1} H(\theta_n,X_{n+1})$; a CLT for theassociated average sequence is also established. The originality of this paperis to address the case of controlled Markov chain dynamics $\{X_n, n\geq 0 \}$and the case of multiple targets. The framework also accomodates (randomly)truncated SA algorithms. Sufficient conditions for CLT's to hold are providedas well as comments on how these conditions extend previous works (such asindependent and identically distributed dynamics, the Robbins-Monro dynamic orthe single target case). The paper gives a special emphasis on how theseconditions hold for SA with controlled Markov chain dynamics and multipletargets; it is proved that this paper improves on existing works.

Central Limit Theorems for Stochastic Approximation with controlled Markov chain dynamics