Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T12:27:13.557Z Has data issue: false hasContentIssue false
  • English
  • Français

Functional and random central limit theorems for the Robbins-Munro process

Published online by Cambridge University Press:  14 July 2016

D. L. McLeish
D. L. McLeish*
Affiliation:
York University, Downsview, Ontario
Get access Rights & Permissions [Opens in a new window]

Abstract

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.

Keywords

DONSKER'S THEOREMINVARIANCE PRINCIPLESTOCHASTIC APPROXIMATIONROBBINS–MUNRO PROCEDUREMARTINGALECENTRAL LIMIT THEOREM
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 1 , March 1976 , pp. 148 - 154
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Blum, J.R. (1954) Approximation methods which converge with probability 1. Ann. Math. Statist. 25, 382–386.Google Scholar
Chung, K. L. (1954) On a stochastic approximation method. Ann. Math. Statist. 25, 463–483.Google Scholar
Fabian, V. (1968) On asymptotic normality in stochastic approximation. Ann. Math. Statist. 39, 1327–1332.Google Scholar
Major, P. and Révész, P. (1973) A limit theorem for the Robbins–Munro approximation. Z. Wahrscheinlichkeitsth. 27, 79–86 CrossRefGoogle Scholar
McLeish, D. L. (1974) Dependent central limit theorems and invariance principles. Ann. Prob. 2.Google Scholar
Rényi, A. (1958) On mixing sequences of random variables. Acta Math. Acad. Scient. Hung. 9, 389–393.Google Scholar
Robbins, H. and Munro, S. (1952) A stochastic approximation method. Ann. Math. Statist. 22, 400–407 CrossRefGoogle Scholar
Sacks, J. (1958) Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist. 29, 373–405.CrossRefGoogle Scholar
Venter, J. H. (1967) An extension of the Robbins–Munro procedure. Ann. Math. Statist. 38, 181–190.CrossRefGoogle Scholar
Vervaat, W. (1972) Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrscheinlichkeitsth. 23, 245–253.Google Scholar
Whitt, W. (1970) Weak convergence of probability measures on the function space C[0, 8). Ann. Math. Statist. 41, 939–944.Google Scholar
Heyde, C. C. (1974) On martingale limit theory and strong convergence results for stochastic approximation procedures. Stoch. Proc. Appl. 2, 359–370.Google Scholar
Stone, C. (1963) Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Amer. Math. Soc. 14, 694–696.CrossRefGoogle Scholar
Venter, J. H. (1966) On Dvoretsky stochastic approximation theorems. Ann. Math. Statist. 37, 1534–1544.Google Scholar