Asymptotic Normality for a Vector Stochastic Difference Equation with Applications in Stochastic Approximation

In this paper, we consider an asymptotic normality problem for a vector stochastic difference equation of the formUn+1=(I+an(B+En)) Un+an(un+en), whereBis a stable matrix, andEn→n0,anis a positive real step size sequence withan→n0, ∑∞n=1 an=∞, anda−1n+1−a−1n→nλ⩾0,unis an infinite-term moving average process, and[formula]. Obviously,anhere is a quite general step size sequence and includes (log n)β/nα, 12<α<1, orα=1 withβ⩾0 as special cases. It is well known that the problem of an asymptotic normality for a vector stochastic approximation algorithm is usually reduced to the above problem. We prove that[formula]converges in distribution to a zero mean normal random vector with covariance ∫∞0 e(B+(1/2) λI) tRe(Bτ+(1/2) λI) tdt, where matrixRdepends only on some stochastic properties ofun, which implies that the asymptotic distributions for both the vector stochastic difference equation and vector stochastic approximation algorithm do not depend on the specific choices ofandirectly but onλ, the limit ofa−1n+1−a−1n.