 
Summary: Convergence of stochastic approximation for Lyapunov stable
dynamics: a proof from ...rst principles.
Christophe Andrieu1
 Éric Moulines2
 Stanislav Volkov1
1
School of Mathematics, University of Bristol
2
ENST, Paris.
July 13, 2004
Abstract
In this short note we present a proof, aimed at beginners, of the convergence of the stochastic
approximation recursion i+1 = i + i+1h( i)+ i+1 i+1 under the classical 0level KushnerClark noise
condition when the underlying dynamic is Lyapunov stable. The technique of proof relies on simple
calculus arguments and bypasses the need for the introduction of the associated continuous time ODE.
Future work includes the extension of the result to the case where the 0level KushnerClark condition
is replaced with a rlevel condition for r > 0 [2].
We study the convergence of the recursion
i+1 = i + i+1h( i) + i+1 i+1;
for a given function h : ! Rn
