Asymptotic Normality for a Vector Stochastic Difference Equation with Applications in Stochastic Approximation*
journal of multivariate analysis 57, 101 118 (1996)，-0001，（）：
In this paper, we consider an asymptotic normality problem for a vector stochastic difference equation of the form Un+1=(I+an(B+En)) Un+an(un+en), where B is a stable matrix, and En→n0, an is a positive real step size sequence with an→n0, Σ∞=1 an=∞, and a-1-a-1→λ≥0, un is an infinite-term moving average process, and en=o(an). Obviously, an here is a quite general step size sequence and includes (log n)β\nα, 1/2<α<1, α=1 with β≥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 Un/√an converges in distribution to a zero mean normal random vector with covariance ∞e (B+(182)λI) tRe(Bt+(1/2)λI)t dt, where matrix R depends only on some stochastic properties of un, 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 of an directly but on λ, the limit of a-1 a-1.
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