Summary
LetB 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
(ϱ and η) positive real numbers,U k, Vk, Tk B-valued random variables, Γ k random linear operatorsB→B, where — in addition to some further assumptions —Γ k →Γ almost surely andT k→T, say, almost surely. Letting (Y k,n )k∈{1,...,n},n∈N be an array ofB-valued random variables such that the partial sums\(\sum\limits_{k = 1}^m {V_k }\) and\(\sum\limits_{k = 1}^m {Y_{k,n} }\) do not differ “too much” (uniformly inm∈{1,...,n}), we shall investigate the distance of (U m ) m∈{1,...,n} and (U m (n) ) m∈{1,...,n}, where
fork∈{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).
References
Abdelhamid, S.N.: Transformation of observations in stochastic approximation. Ann. Statist.1, 1158–1174 (1973)
Berger, E.: A note on the invariance principle for stochastic approximation procedures in a Hilbert space. Unpublished manuscript (1979)
Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968
Birnbaum, Z.W., Marshall, A.W.: Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Ann. Math. Statist.32, 687–704 (1961)
Blum, J.R.: Approximation methods which converge with probability one. Ann. Math. Statist.25, 382–386 (1954)
Borodin, A.N.: A stochastic approximation procedure in the case of weakly dependent observations. Theory Probab. Appl.24 34–52 (1979)
Chung, K.L.: On a stochastic approximation method. Ann. Math. Statist.25, 463–483 (1954)
Daleckiî, Ju.L., Kreîn, M.G.: Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs43. Providence, Rhode Island: American Math. Society 1974
Dehling, H.: Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.63, 393–432 (1983)
Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Statist.39, 1563–1572 (1968)
Dudley, R.M., Philipp, W.: Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 509–552 (1983)
Fabian, V.: On asymptotic normality in stochastic approximation. Ann. Math. Statist.39, 1327–1332 (1968)
Fabian, V.: Asymptotically efficient stochastic approximation; the RM case. Ann. Statist.1, 485–495 (1973)
Fabian, V.: On asymptotically efficient recursive estimation. Ann. Statist.6, 854–866 (1978)
Fabian, V.: Local asymptotic minimax properties of some recursive estimates.RM — 409. Department of Statistics and Probability. Michigan State University (1980)
Fernique, X.: Intégrabilité des vecteurs gaussiens. C.R. Acad. Sci. Paris Sér.A 270, A 1698–1699 (1970)
Gänssler, P., Stute, W.: Wahrscheinlichkeitstheorie. Berlin-Heidelberg-New York: Springer 1977
Gaposhkin, V.F., Krasulina, T.P.: On the law of the iterated logarithm in stochastic approximation processes. Theory Probab. Appl.19, 844–850 (1974)
Hall, P.G., Heyde, C.C.: On a unified approach to the law of the iterated logarithm for martingales. Bull. Aust. Math. Soc.14, 435–447 (1976)
Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Applications. New York-San Francisco-London: Academic Press 1980
Heyde, C.C.: On martingale limit theory and strong convergence results for stochastic approximation procedures. Stochastic Processes Appl.2, 371–389 (1974)
Jakubowski, A.: On limit theorems for sums of dependent Hilbert space valued random variables. In: Mathematical Statistics and Probability Theory. Proc. 6th Int. Conf., Wisla/Pol. 1978. Lect. Notes Stat.2, 178–187 (1980)
Kersting, G.: Almost sure approximation of the Robbins-Monro process by sums of independent random variables. Ann. Probab.5, 954–965 (1977)
Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheor. Verw. Geb.32, 111–131 (1975)
Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV's and the sample DF. II. Z. Wahrscheinlichkeitstheor. Verw. Geb.34, 33–58 (1976)
Kuelbs, J., LePage, R.: The law of the iterated logarithm for Brownian motion in a Banach space. Trans. Am. Math. Soc.185, 253–264 (1973)
Kuelbs, J., Philipp, W.: Almost sure invariance principles for partial sums of mixing B-valued random variables. Ann. Probab.8, 1003–1036 (1980)
Lai, T.L., Robbins, H.: Limit theorems for weighted sums and stochastic approximation processes. Proc. Nat. Acad. Sci. U.S.A.75, 1068–1070 (1978)
Lai, T.L., Robbins, H.: Adaptive design and stochastic approximation. Ann. Statist.7, 1196–1221 (1979)
Major, P., Révész, P.: A limit theorem for the Robbins-Monro approximation. Z. Wahrscheinlichkeitstheor. Verw. Geb.27, 79–86 (1973)
Major, P.: The approximation of partial sums of independent RV's Z. Wahrscheinlichkeitstheor. Verw. Geb.35, 213–220 (1976)
Mark, G.: Log-log-Invarianzprinzipien für Prozesse der stochastischen Approximation. Mitteilungen Math. Sem. Giessen, No. 153, 87 pp. (1982)
McLeish, D.L.: Invariance principles for dependent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.32, 165–178 (1975)
McLeish, D.L.: Functional and random central limit theorems for the Robbins-Monro process. J. Appl. Probab.13, 148–154 (1976)
McLeish, D.L.: On the invariance principle for nonstationary mixingales. Ann. Probab.5, 616–621 (1977)
Métivier, M.: Semimartingales: a Course on Stochastic Processes. Berlin-New York: de Gruyter 1982
Morrow, G., Philipp, W.: An almost sure invariance principle for Hilbert space valued martingales. Trans. Am. Math. Soc.273, 231–251 (1982)
Nevel'son, M.B., Has'minskiî, R.Z.: Stochastic Approximation and Recursive Estimation. Translations of Mathematical Monographs47. Providence, Rhode Island: American Math. Society 1976
Nixdorf, R.: Stochastische Approximation in Hilberträumen durch endlichdimensionale Verfahren. Mitteilungen Math. Sem. Giessen, No. 154, 78 pp. (1982)
Nixdorf, R.: An invariance principle for a finite dimensional stochastic approximation method in a Hilbert space. J. Multivariate Analysis15, 252–260 (1984)
O'Reilly, N.E.: On weak convergence of empirical processes in sup-norm metrics. Ann. Probab.2, 642–651 (1974)
Pantel, M.: Adaptive Verfahren der stochastischen Approximation. Dissertation. Universität Essen — Gesamthochschule (1979)
Philipp, W., Stout, W.: Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. No. 161. Providence, Rhode Island: American Math. Society 1975
Philipp, W.: Almost sure invariance principles for sums ofB-valued random variables. In: Probability in Banach Spaces II. Proc. Conf. Oberwolfach 1978. Lect. Notes in Math.709, pp. 171–193, Berlin-Heidelberg-New York: Springer 1979
Pyke, R.: Applications of almost surely convergent constructions of weakly convergent processes. In: Probability and Information Theory. Proc. Int. Symp. McMaster Univ. 1968. Lect. Notes in Math.89, 187–200. Berlin-Heidelberg-New York: Springer 1969
Ruppert, D.: Almost sure approximations to the Robbins-Monro and Kiefer-Wolfowitz processes with dependent noise. Ann. Probab.10, 178–187 (1982)
Sacks, J.: Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist.29, 373–405 (1958)
Skorohod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl.1, 261–290 (1956)
Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 211–226 (1964)
Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Sympos. Math. Statist. Probab.2, 315–344 (1965)
Venter, J.H.: An extension of the Robbins-Monro procedure. Ann. Math. Statist.38, 181–190 (1967)
Walk, H.: An invariance principle for the Robbins-Monro process in a Hilbert space. Z. Wahrscheinlichkeitstheor. Verw. Geb.39, 135–150 (1977)
Walk, H.: Sequential estimation of the solution of an integral equation in filtering theory. In: Stochastic Control Theory and Stochastic Differential Systems (eds.: M. Kohlmann, W. Vogel). Lect. Notes in Control and Information Sciences16, 598–605 (1979)
Walk, H.: Martingales and the Robbins-Monro procedure inD[0, 1].J. Multivariate Anal. 8, 430–452 (1978)
Walk, H.: A functional central limit theorem for martingales inC(K) and its application to sequential estimates. J. Reine Angew. Math.314, 117–135 (1980)
Watanabe, M.: A stochastic approximation from dependent observations. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 279–292 (1983)
Author information
Authors and Affiliations
Additional information
Part of this work was done while the author was supported by Deutsche Forschungsgemeinschaft
Rights and permissions
About this article
Cite this article
Berger, E. Asymptotic behaviour of a class of stochastic approximation procedures. Probab. Th. Rel. Fields 71, 517–552 (1986). https://doi.org/10.1007/BF00699040
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00699040
Keywords
- Real Number
- Stochastic Process
- Asymptotic Behaviour
- Probability Theory
- Limit Theorem