Stochastic Approximations via Large Deviations: Asymptotic Properties

1985 ◽  
Vol 23 (5) ◽  
pp. 675-696 ◽  
Author(s):  
Paul Dupuis ◽  
Harold J. Kushner
Keyword(s):  
Large Deviations ◽  
Asymptotic Properties ◽  
Stochastic Approximations

Self-normalized asymptotic properties for the parameter estimation in fractional Ornstein–Uhlenbeck process

2019 ◽  
Vol 19 (03) ◽  
pp. 1950018 ◽  
Author(s):  
Hui Jiang ◽  
Junfeng Liu ◽  
Shaochen Wang
Keyword(s):  
Large Deviations ◽  
Asymptotic Properties ◽  
Moderate Deviations ◽  
Delta Method ◽  
Fourth Moment ◽  
Test Statistics ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Deviation Inequalities ◽  
Type Ii Errors

In this paper, we consider the self-normalized asymptotic properties of the parameter estimators in the fractional Ornstein–Uhlenbeck process. The deviation inequalities, Cramér-type moderate deviations and Berry–Esseen bounds are obtained. The main methods include the deviation inequalities and moderate deviations for multiple Wiener–Itô integrals [P. Major, Tail behavior of multiple integrals and U-statistics, Probab. Surv. 2 (2005) 448–505; On a multivariate version of Bernsteins inequality, Electron. J. Probab. 12 (2007) 966–988; M. Schulte and C. Thäle, Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem, J. Funct. Anal. 270 (2016) 2223–2248], as well as the Delta methods in large deviations [F. Q. Gao and X. Q. Zhao, Delta method in large deviations and moderate deviations for estimators, Ann. Statist. 39 (2011) 1211–1240]. For applications, we propose two test statistics which can be used to construct confidence intervals and rejection regions in the hypothesis testing for the drift coefficient. It is shown that the Type II errors tend to be zero exponential when using the proposed test statistics.

scholarly journals Cramér type large deviations for trimmed L-statistics

2018 ◽  
Vol 37 (1) ◽  
pp. 101-118 ◽  
Author(s):  
Nadezhda Gribkova
Keyword(s):  
Large Deviations ◽  
Weight Function ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Natural Conditions ◽  
New Approach ◽  
Probabilities Of Large Deviations ◽  
Large Deviation Problem ◽  
Deviation Problem

CRAMÉR TYPE LARGE DEVIATIONS FOR TRIMMED L-STATISTICSIn this paper, we propose a new approach to the investigationof asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. 1982 – the first and, as far as we know, the single article where some results on probabilities of large deviations for the trimmed L-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed L-statistic by a non-trimmed L-statistic with smooth weight function based onWinsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.

scholarly journals Large deviations for Lotka-Nagaev estimator of a randomly indexed branching process

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5803-5808 ◽  
Author(s):  
Zhenlong Gao ◽  
Lina Qiu
Keyword(s):  
Large Deviations ◽  
Renewal Process ◽  
Continuous Time ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Time Process ◽  
Continuous Time Process ◽  
Watson Process

Consider a continuous time process {Yt=ZNt, t ? 0}, where {Zn} is a supercritical Galton-Watson process and {Nt} is a renewal process which is independent of {Zn}. Firstly, we study the asymptotic properties of the harmonic moments E(Y-rt) of order r > 0 as t ? ?. Then, we obtain the large deviations of the Lotka-Negaev estimator of offspring mean.

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