A Newton-Raphson version of the multivariate Robbins-Monro procedure (Q1061434)

To find the unique \(\theta\) such that \(f(\theta)=0\), \(f:R^ k\to R^ k,\) the Robbins-Monro procedure \(X_{n+1}=X_ n-an^{-1}B_ nD_ nf_ n\) is introduced, where \(f_ n\) is an estimate of \(f(X_ n)\), \(D_ n\) is an estimate of the derivative \(D(X_ n)\) of f at \(X_ n\), \(B_ n\) is an estimate of \([D^ t(\theta)D(\theta)]^{-1}\), and a is a constant. The basic result is that \(f(X_ n)\to 0\) a.s., and under additional assumptions a.s. convergence of \(X_ n\) to \(\theta\) and asymptotic normality of \(X_ n\) are established.