Arithmetic means and invariance principles in stochastic approximation (Q1209213)

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.