Exact bounds for the rate of convergence in general stochastic approximation procedures

References

• 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 .
   Web of Science ®Google Scholar
• Dupuis , P. and Kushner , H.J. 1989 . Stochastic approximation and large deviations: upper bounds and w.p. convergence . SIAM J. Control Optimization , 27 : 1108 – 1135 .
   Web of Science ®Google Scholar
• Gaposhkin , V.F. and Krasulina , T.P. 1974 . On the law of the iterated logarithm in stochastic approximation processes . Theory Probab. Appl , 19 : 844 – 850 .
   Google Scholar
• Heyde , C.C. 1974 . On martingale limit theory and strong convergence results for stochastic approximation procedures . Stochastic Processes Appl , 2 : 371 – 389 .
   Google Scholar
• Kersting , G. 1977 . Almost sure approximation of the Robbins-Monro process by sums of independent random variables . Ann. Probab , 5 : 954 – 965 .
   Google Scholar
• Klesov , O.I. 1987 . The law of the iterated logarithm for weighted sums of independent identically distributed random variables . Theory Probab. Appl , 31 : 337 – 342 .
   Google Scholar
• Komlós , J. , Major , P. and Tusnády , G. 1975 . An approximation of partial sums of independent RV's and the sample DF I II . Z. Wahrscheinlichkeitstheorie verw. Geb , 32 : 111 – 131 . 34 (1976) 33-58
   Web of Science ®Google Scholar
• Korostelev , A.P. 1983 . On upper functions for stochastic approximation procedures . Theory Probab. Appl , 28 : 806 – 811 .
   Google Scholar
• Korostelev , A.P. 1984 . Stochastic recursive procedures (local properties) , Moscow : Nauka . in Russian
   Google Scholar
• Koval , V.A. 1994 . On the rate of convergence for stochastic approximation procedures . Ukrain. Math. Journ , 46 : 1093 – 1100 .
   Google Scholar
• Li , G. 1994 . Almost sure convergence of stochastic approximation procedures . Statist. Sinica , 4 : 361 – 372 .
   Google Scholar
• Ljung , L. 1974 . Convergence of recursive stochastic, algorithms , Lund Institute of Technology . Report 7403, Dept. of Automatic Control
   Google Scholar
• Ljung , L. 1978 . Strong convergence of a stochastic approximation algorithm . Ann. Statist , 6 : 680 – 696 .
   Web of Science ®Google Scholar
• Major , P. 1976 . The approximation of partial sums of independent rv's . Z. Wahrscheinlichkeitstheorie veriu. Geb , 35 : 213 – 220 .
   Google Scholar
• Mark , G. 1982 . Loglog-Invarianzprinzipien für Prozesse der stochastischen Approximation . Mitteilungen Math. Sem. Gieβen , 153
   Google Scholar
• Petrov , V.V. 1995 . Limit theorems of probability theory: sequences of independent random variables , Oxford : Clarendon .
   Google Scholar
• Philipp , W. 1986 . “ Invariance principles for independent and weakly dependent random variables ” . In Dependence in Probability and Statistics. Progress in Probability and Statistics , Edited by: Eberlein , E. and Taqqu , M.S. Vol. 11 , 225 – 268 . Basel : Birkliäuser .
   Google Scholar
• Philipp , W. and Stout , W. 1975 . Almost sure invariance principles for partial sums of weakly dependent random variables . Mem. Am. Math. Soc , 161
   Google Scholar
• Rio , E. 1995 . The functional law of the iterated logarithm for stationary strongly mixing sequences . Ann. Probab , 23 : 1188 – 1203 .
   Web of Science ®Google Scholar
• Ruppert , D. 1982 . Almost sure approximation to the Robbins-Monro and Kiefer-Wolfowitz processes with dependent noise . Ann. Probab , 10 : 178 – 187 .
   Web of Science ®Google Scholar
• Schwabe , R. and Walk , H. 1996 . On a stochastic approximation procedure based on averaging . Metrika ,
   Google Scholar
• Shao , Q.M. 1987 . Strong approximations on lacunary trigonometric series with weights . Sci. Sinica(A) , 30 : 796 – 806 .
   Google Scholar
• Shao , Q.M. 1993 . Almost sure invariance principles for mixing sequences of random variables . Stochastic Processes Appl , 48 : 319 – 334 .
   Web of Science ®Google Scholar
• Shao , Q.M. 1995 . Strong approximation theorems for independent random variables and their applications . J. Multivariate Anal , 52 : 107 – 130 .
   Google Scholar
• Shao , Q.M. and Lu , C.R. 1987 . Strong approximations for partial sums of weakly dependent random variables . Sci. Sinica (A) , 30 : 575 – 587 .
   Google Scholar
• Solo , V. 1982 . Stochastic approximation and the final value theorem . Stochastic Processes Appl , 13 : 139 – 156 .
   Google Scholar
• Stout , W. 1974 . Almost sure convergence , New York : Academic Press .
   Google Scholar
• Walk , H. 1988 . Limit behaviour of stochastic approximation processes . Statist. Decis , 6 : 109 – 128 .
   Google Scholar
• Walk H. Foundations of stochastic approximation Approximation and Optimization of Random Systems DMV Seminar Blaubeuren 1989 DMV Seminar 17. Birkhäuser Basel May 28-June 4 1992 1 51 In: L. Ljung, G. Pflug and H. Walk
   Google Scholar
• Walk , H. and Zsidó , L. 1989 . Convergence of the Robbins-Monro method for linear problems in a Banach space . J. Math. Anal. Appl , 139 : 152 – 177 .
   Web of Science ®Google Scholar