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

✍ Scribed by Yunmin Zhu

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 651 KB
• Volume: 57
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1996.0024

✦ Synopsis

In this paper, we consider an asymptotic normality problem for a vector stochastic difference equation of the form

where B is a stable matrix, and E n Ä n 0, a n is a positive real step size sequence with a n Ä n 0, n=1 a n = , and a &1 n+1 &a &1 n Ä n * 0, u n is an infinite-term moving average process, and e n =o(a n ). Obviously, a n here is a quite general step size sequence and includes (log n) ; Ân : , 1 2 <:<1, or :=1 with ; 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 U n Â-a n converges in distribution to a zero mean normal random vector with covariance 0 e (B+(1Â2) *I ) t Re (B { +(1Â2) *I ) t dt, where matrix R depends only on some stochastic properties of u n , 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 a n directly but on *, the limit of a &1 n+1 &a &1 n .