Convergence of Markovian Stochastic Approximation with discontinuous dynamics

This paper is devoted to the convergence analysis of stochastic approximation algorithms of the form $\theta\_{n+1} = \theta\_n + \gamma\_{n+1} H\_{\theta\_n}(X\_{n+1})$ where $\{\theta\_nn, n \geq 0\}$ is a $R^d$-valued sequence, $\{\gamma, n \geq 0\}$ is a deterministic step-size sequence and $\{X\_n, n \geq 0\}$ is a controlled Markov chain. We study the convergence under weak assumptions on smoothness-in-$\theta$ of the function $\theta \mapsto H\_{\theta}(x)$. It is usually assumed that this function is continuous for any $x$; in this work, we relax this condition. Our results are illustrated by considering stochastic approximation algorithms for (adaptive) quantile estimation and a penalized version of the vector quantization.