Geodesic
Central limit theorems for stochastic approximation with controlled Markov chain dynamics
Fort, Gersende1 
1 LTCI, CNRS & TELECOM ParisTech, 46 rue Barrault, 75634 Paris cedex 13, France
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 60-80

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This paper provides a Central Limit Theorem (CLT) for a process {θ n ,n≥0} satisfying a stochastic approximation (SA) equation of the form θ n+1 =θ n +γ n+1 H(θ n ,X n+1 ); a CLT for the associated average sequence is also established. The originality of this paper is to address the case of controlled Markov chain dynamics {X n ,n≥0} and the case of multiple targets. The framework also accomodates (randomly) truncated SA algorithms. Sufficient conditions for CLT’s hold are provided as well as comments on how these conditions extend previous works (such as independent and identically distributed dynamics, the Robbins−Monro dynamic or the single target case). The paper gives a special emphasis on how these conditions hold for SA with controlled Markov chain dynamics and multiple targets; it is proved that this paper improves on existing works.

Reçu le :
DOI : 10.1051/ps/2014013
Classification : 62L20, 60F05, 60J20
Keywords: Stochastic approximation, limit theorems, controlled markov chain

Fort, Gersende 1

1 LTCI, CNRS & TELECOM ParisTech, 46 rue Barrault, 75634 Paris cedex 13, France 
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     author = {Fort, Gersende},
     title = {Central limit theorems for stochastic approximation with controlled {Markov} chain dynamics},
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     publisher = {EDP-Sciences},
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Fort, Gersende. Central limit theorems for stochastic approximation with controlled Markov chain dynamics. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 60-80. doi: 10.1051/ps/2014013

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