Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T07:44:37.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behavior for the Robbins–Monro process

Part of: Parametric inference Limit theorems

Published online by Cambridge University Press:  26 July 2018

Yu Miao  and
Manru Dong
Yu Miao*
Affiliation:
Henan Normal University
Manru Dong*
Affiliation:
Henan Normal University
*
* Postal address: College of Mathematics and Information Science, Henan Normal University, 46# East of Construction Road, Xinxiang, Henan, 453007, China.
* Postal address: College of Mathematics and Information Science, Henan Normal University, 46# East of Construction Road, Xinxiang, Henan, 453007, China.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the Robbins–Monro procedure Xn+1 = Xn - an-1Yn with some fixed number a > 0 and establish the moderate deviation principle of the process {Xn}.

Keywords

Robbins–Monro processmoderate deviationexponential inequality

MSC classification

Primary: 62F12: Asymptotic properties of estimators
Secondary: 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 559 - 570
Copyright
Copyright © Applied Probability Trust 2018 

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

[1]Arab, I. and Dahmani, A. (2016). Complete convergence of stochastic approximation algorithm in R d under random noises. Sequential Anal. 35, 216–225. Google Scholar
[2]Blum, J. R. (1954). Approximation methods which converge with probability one. Ann. Math. Statist. 25, 382–386. Google Scholar
[3]Burkholder, D. L. (1956). On a class of stochastic approximation processes. Ann. Math. Statist. 27, 1044–1059. Google Scholar
[4]Chung, K. L. (1954). On a stochastic approximation method. Ann. Math. Statist. 25, 463–483. Google Scholar
[5]Dahmani, A. and Ait Saidi, A. (2010). Consistency of Robbins Monro's algorithm within a mixing framework. Bull. Soc. R. Sci. Liège 79, 131–140. Google Scholar
[6]Derman, C. (1956). An application of Chung's lemma to the Kiefer-Wolfowitz stochastic approximation procedure. Ann. Math. Statist. 27, 532–536. Google Scholar
[7]Dippon, J. (2004). Higher order representations of the Robbins–Monro process. J. Multivariate Anal. 90, 301–326. Google Scholar
[8]Gapoškin, V. F. and Krasulina, T. P. (1975). The law of the iterated logarithm in stochastic approximation processes. Theory Prob. Appl. 19, 844–850. Google Scholar
[9]HodgesJ. L., Jr. J. L., Jr. and Lehmann, E. L. (1956). Two approximations to the Robbins–Monro process. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. I, University of California Press, Berkeley, pp. 95–104. Google Scholar
[10]Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30. Google Scholar
[11]Kersting, G. D. (1978). A weak convergence theorem with application to the Robbins–Monro process. Ann. Prob. 6, 1015–1025. Google Scholar
[12]Major, P. and Révész, P. (1973). A limit theorem for the Robbins–Monro approximation. Z. Wahrscheinlichkeitsth. 27, 79–86. Google Scholar
[13]Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22, 400–407. Google Scholar
[14]Sacks, J. (1958). Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist. 29, 373–405. Google Scholar
[15]Walk, H. (1977). An invariance principle for the Robbins-Monro process in a Hilbert space. Z. Wahrscheinlichkeitsth. 39, 135–150. Google Scholar
[16]Woodroofe, M. (1972). Normal approximation and large deviations for the Robbins-Monro process. Z. Wahrscheinlichkeitsth. 21, 329–338. Google Scholar