Rate of mixing and the averaging principles for stochastic recursive procedures (Q1177622)

This paper considers the stochastic algorithm \[ \vartheta_{n+1}=\vartheta_ n+\gamma f(\vartheta_ n,Y_{n+1}),\tag{1} \] \[ Y_{n+1}=Y_ n+a_ n(\vartheta_ n,Y_ n)+b_ n(\vartheta,Y_ n)W_ n.\tag{2} \] The authors first derive an estimate for the rate of mixing for the process (2) when \(\vartheta_ n\) is held fixed to the value \(\vartheta\), under quite general conditions on the distribution of the i.i.d. sequence \(\{W_ n\}\). From this they prove the convergence of \(\vartheta_ n\), suitably scaled to a jump process with interval \(\gamma\) between successive jumps, to the solution of the averaged ODE as \(\gamma\) tends to 0, \(\dot\vartheta(t)=h(\vartheta(t))\), \(h(\vartheta)\) being the mean of \(f(\vartheta,Y)\) under the stationary measure for the process frozen to \(\vartheta\).