On moderate deviations for martingales
type: Item
instance of: scholarly article
title: On moderate deviations for martingales
zbMATH Open document ID: 0881.60026
author: Ion Grama
published in: The Annals of Probability
publication date: 1998-01-26T00:00:00Z
review text: The paper considers square-integrable martingales \((X_t\), \(0\leq t\leq1)\) and investigates the large deviation problem \(P(X_1\geq r)=(1-\Phi(r))(1+\theta(r)R)\) for each \(|r|\leq C\), where \(|\theta(r)|\leq1\). The author receives a precise description of the remainder \(R\) and of the range \(C\) by means of \(L_{2\delta}=E\;\sum_{0<s\leq1}|\Delta X_s|^{2\delta}\) and \(N_{2\delta}=E|\langle X\rangle_1-1|^{1+\delta}\), where \(\Delta X\) is the jumping part and \(\langle X\rangle\) is the quadratic characteristic of the martingale \(X\). The result gives a rate for large deviations in martingale CLT and even improves the result known for the case of independent random variables.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 62E17
zbMATH DE Number: 1018385
zbMATH Keywords: martingale
zbMATH Keywords: central limit theorem
zbMATH Keywords: rate of convergence
zbMATH Keywords: moderate deviation
reviewed by: Petr Lachout
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1024404283
OpenAlex ID: W1987784253
cites work: Q5836164
cites work: Q3761357
cites work: On probabilities of large deviations in Banach spaces
cites work: Exact convergence rates in some martingale central limit theorems
cites work: On the Departure from Normality of a Certain Class of Martingales
cites work: Asymptotic normality of MRPP statistics from invariance principles of u-statistics
cites work: On the rate of convergence in the central limit theorem for d-dimensional semimartingales
cites work: The probabilities of large deviations for semimartingales
cites work: On the rate of convergence in the central limit theorem for martingales with discrete and continuous time
cites work: A nonuniform bound on the rate of convergence in the martingale central limit theorem
cites work: Q5535438
cites work: Q3774629
cites work: On the Rate of Convergence in the Central Limit Theorem for Semimartingales
cites work: Q3995465
cites work: Q4776687
cites work: Q5568451
DOI: 10.1214/AOP/1024404283
description: scientific article; zbMATH DE number 1018385