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A Newton-Raphson version of the multivariate Robbins-Monro procedure. (English) Zbl 0571.62072
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.
Reviewer: R.Zielinski

##### MSC:
 62L20 Stochastic approximation 60F15 Strong limit theorems 60F05 Central limit and other weak theorems
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