Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 62L20 Stochastic approximation
- 60J10 Markov chains with discrete parameter
- 90C15 Stochastic programming

**Résumé:**
In this paper we address the problem of the stability of the
stochastic approximation procedure. The stability of such
algorithms is known to rely heavily on the growth of the mean
field at the boundary of the parameter set and the magnitude of
the sizesteps used in the procedure. The conditions typically
required to ensure convergence are either too difficult to check
in practice or not satisfied at all, even for simple models. The
most popular technique to circumvent this problem consists of
constraining the parameter to a compact subset in the parameter
space.
We propose and analyze here an alternative, based on projection on
adaptive truncation sets, extending previous works in this
direction. This procedure allows for the adaptive tuning of the
magnitude of the stepsizes, which is key to ensuring stability.
The stability - with probability one - of the scheme is proved
under a set of verifiable assumptions. We illustrate these claims
in the so-called controlled Markovian setting and present two
substantial examples. The first example is related to the minimum
prediction error estimation of the parameters of stable and
invertible ARMA processes and the second example is related to
controlled Markov chain Monte Carlo algorithms.

**Mots Clés:** *Stochastic approximation ; state-dependent noise ; randomly varying
truncation ; Adaptive Markov Chain Monte Carlo*

**Date:** 2003-02-10

**Prépublication numéro:** *PMA-791*

**Postscript file: **PMA-791.ps