Asymptotic behaviour of a class of stochastic approximation procedures
Let \({\mathbb{B}}\) be a real Banach space. Considering stochastic approximation procedures of Robbins-Monro and Kiefer-Wolfowitz type, one is led to recursion formulas of the form \[ U_{k+1}=U_ k-k^{- 1}\Gamma_ kU_ k-k^{-\rho}V_ k-k^{-\eta}T_ k \] \(\rho\) and \(\eta\) positive real numbers, \(U_ k\), \(V_ k\), \(T_ k\) \({\mathbb{B}}\)- valued random variables, \(\Gamma_ k\) random linear operators \({\mathbb{B}}\to {\mathbb{B}}\), where - in addition to some further assumptions - \(\Gamma_ k\to \Gamma\) almost surely and \(T_ k\to T\), say, almost surely. Letting \((Y_{k,n})_{k\in \{1,...,n\}}\), \(n\in {\mathbb{N}}\), be an array of \({\mathbb{B}}\)-valued random variables such that the partial sums \(\sum^{m}_{k=1}V_ k\) and \(\sum^{m}_{k=1}Y_{k,n}\) do not differ ''too much'' (uniformly in \(m\in \{1,...,n\})\), we shall investigate the distance of \((U_ m)_{m\in \{1,...,n\}}\) and \((U_ m^{(n)})_{m\in \{1,...,n\}}\), where \[ U^{(n)}_{k+1}:=U_ k^{(n)}-k^{-1}\Gamma U_ k^{(n)}-k^{-\rho}Y_{k,n}-k^{- \eta}T,\quad k\in \{1,...,n-1\}. \] The proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems, weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence \((U_ n)\).
weak and almost sure invariance principles
variants of the bounded and the functional law of the iterated logarithm
scientific article; zbMATH DE number 3911528
scientific article