An almost sure invariance principle for stochastic approximation procedures in linear filtering theory. (English) Zbl 0877.62078

Summary: We consider a class of stochastic approximation procedures that arises in linear filtering and regression theory. Our main result asserts that the stochastic approximation process satisfies an almost sure invariance principle (with a certain rate of convergence) if the partial sums of the errors do.


62L20 Stochastic approximation
60F17 Functional limit theorems; invariance principles
62M20 Inference from stochastic processes and prediction
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[1] Berger, E. (1986). Asy mptotic behaviour of a class of stochastic approximation procedures. Probab. Theory Related Fields 71 517-552. · Zbl 0571.62073
[2] Berger, E. (1990). An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables. Probab. Theory Related Fields 84 161-201. · Zbl 0695.60041
[3] Dalecki i, Ju. L. and Kre in, M. G. (1974). Stability of Solutions of Differential Equations in Banach Space. Amer. Math. Soc., Providence, RI.
[4] Fabian, V. (1968). On asy mptotic normality in stochastic approximation. Ann. Math. Statist. 39 1327-1332. · Zbl 0176.48402
[5] Heunis, A. (1994). Rates of convergence for an adaptive filtering algorithm driven by stationary dependent data. SIAM J. Control Optim. 32 116-139. · Zbl 0789.60027
[6] Kouritzin, M. A. (1996). On the interrelation of almost sure invariance principles for certain stochastic adaptive algorithms and for partial sums of random variables. J. Theoret. Probab. 9 811-840. · Zbl 0878.60029
[7] Mark, G. (1982). Log-log-Invarianzprinzipien f ür Prozesse der stochastischen Approximation. Mitt. Math. Sem. Giessen 153 1-87. · Zbl 0486.62084
[8] Philipp, W. (1986). Invariance principles for independent and weakly dependent random variables. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 225-268. Birkhäuser, Boston. · Zbl 0614.60027
[9] Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3 211-226. · Zbl 0132.12903
[10] Walk, H. (1977). An invariance principle for the Robbins-Monro process in a Hilbert space. Z. Wahrsch. Verw. Gebiete 39 135-150. · Zbl 0342.62060
[11] Walk, H. (1980). A functional central limit theorem for martingales in C K and its application to sequential estimates. J. Reine Angew. Math. 314 117-135. · Zbl 0419.60028
[12] Walk, H. (1988). Limit behaviour of stochastic approximation processes. Statist. Decisions 6 109- 128. · Zbl 0668.62058
[13] Walk, H. and Zsid ó, L. (1989). Convergence of the Robbins-Monro method for linear problems in a Banach space. J. Math. Anal. Appl. 139 152-177. · Zbl 0683.62041
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