Arithmetic means and invariance principles in stochastic approximation
type: Item
instance of: scholarly article
title: Arithmetic means and invariance principles in stochastic approximation
zbMATH Open document ID: 0774.62086
DOI: 10.1007/BF01046773
author: Andreas Pechtl
published in: Journal of Theoretical Probability
publication date: 1993-05-16T00:00:00Z
review text: The limit results for the stochastic approximation procedures given by \textit{D. Ruppert} [Tech. Rep. No. 781, School Oper. Res. Ind. Eng., Cornell Univ. (1988)], \textit{B. T. Polyak} [Autom. Remote Control 51, No. 7, 937-946 (1990); translation from Avtom. Telemekh. 1990, No. 7, 98-107 (1990; Zbl 0737.93080)] and \textit{B. T. Polyak} and \textit{A. B. Juditsky} [Tech. Rep. Inst. Contr. Sci., Moscow (1990)] are generalized for the case of a stochastic approximation sequence in a real separable Banach space defined by \[ U_{n+1}=(I-\tau_ n A)U_ n+ \tau_ n V_ n. \] Summation for the sequence \(U_ n\) leads to functional limit theorems if the sequence \(\tau_ n\) is constant or decreases rather slowly to zero and the bounded linear operator \(A\) satisfies a spectral condition. The invariance principle for arithmetic means and the loglog invariance principle are proved for this situation.
reviewed by: T.Fiala
Mathematics Subject Classification ID: 62L20
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 167542
zbMATH Keywords: functional central limit theorems
zbMATH Keywords: real separable Banach space
zbMATH Keywords: invariance principle for arithmetic means
zbMATH Keywords: loglog invariance
MaRDI profile type: Publication
cites work: Q5560061
cites work: New method of stochastic approximation type
cites work: Acceleration of Stochastic Approximation by Averaging
cites work: Q3821437
cites work: Q5531981
description: scientific article; zbMATH DE number 167542