Passive stochastic approximation with constant step size and window width

Motivated by the desire to solve a steady-state estimation and detection problem in chemical engineering, and inspired by the recent progress in passive stochastic approximation, recursive algorithms combining stochastic approximation and kernel estimation are studied and developed in this work. The underlying problem can be stated as finding the roots f(x)=0 provided only noisy measurements y_n=f(x_n,ξ_n) are available, where Ef(x, ξ_n)=f(x). The main difficulty lies in that unlike the traditional approach in stochastic approximation, the sequence {x _n} is generated randomly and cannot be adjusted in accordance with our wish. Similar to those used in the decreasing step size algorithms, another sequence {z_n} is generated to approximate the roots of f¯(x)=0. Some of the features of the algorithms include: constant step size and constant window width and correlated random processes. Under fairly general conditions, it is proven that a weak convergence result holds for an interpolated sequence of the iterates. Error bounds are obtained and a local limit theorem is also derived. The algorithm is then applied to an estimation problem in chemical engineering. Simulation has shown promising results