Exponential inequalities for self-normalized martingales with applications

Abstract

We propose several exponential inequalities for self-normalized martingales similar to those established by De la Peña. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided.

Published in at http://dx.doi.org/10.1214/07-AAP506 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws
• Local limit theory and large deviations for supercritical Branching processes
• Pseudo-maximization and self-normalized processes

Cited by in corpus (32)

• Lasso and probabilistic inequalities for multivariate point processes
• Time-uniform Chernoff bounds via nonnegative supermartingales
• Gaussian Process Optimization with Mutual Information
• Analysis of the Forward Search using some new results for martingales and empirical processes
• Optimal No-regret Learning in Repeated First-price Auctions
• Self-normalized Cramér type moderate deviations for stationary sequences and applications
• Does the Markov Decision Process Fit the Data: Testing for the Markov Property in Sequential Decision Making
• A uniform law for convergence to the local times of linear fractional stable motions
• A central limit theorem for random walk in random environment on marked Galton-Watson trees
• Stochastic Online Optimization using Kalman Recursion
• Multiscale Autoregression on Adaptively Detected Timescales
• Martingale inequalities of type Dzhaparidze and van Zanten
• Majorizing Measures, Sequential Complexities, and Online Learning
• Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion
• Sample Complexity of the Sign-Perturbed Sums Identification Method: Scalar Case
• Bound on the maximal function associated to the bounded law of the iterated logarithms via orthomartingale approximation
• New robust inference for predictive regressions
• Learning via Wasserstein-Based High Probability Generalisation Bounds
• Adaptive wavelet estimation of the diffusion coefficient under additive error measurements
• Bound on the maximal function associated to the law of the iterated logarithms for Bernoulli random fields
• Cramér's moderate deviations for martingales with applications
• Uniform convergence rates over maximal domains in structural nonparametric cointegrating regression
• Deviation inequalities for martingales with applications to linear regressions and weak invariance principles
• Moderate deviations for the mildly stationary autoregressive models with dependent errors
• Logarithmic Regret for parameter-free Online Logistic Regression
• Bernstein type inequalities for self-normalized martingales with applications
• Cramér type moderate deviations for self-normalized $\psi$-mixing sequences
• Large deviation inequalities for martingales in Banach spaces
• Berry--Esseen bounds for self-normalized sums of local dependent random variables
• Moderate deviations for the Durbin-Watson statistic related to the first-order autoregressive process
• Dzhaparidze-van Zanten type inequalities for self-normalized martingales
• Nonuniform Berry-Esseen bounds for martingales with applications to statistical estimation