Related papers: Sharp Finite-Time Iterated-Logarithm Martingale Co…
We develop a new framework for deriving time-uniform concentration bounds for the output of stochastic sequential algorithms satisfying certain recursive inequalities akin to those defining the almost-supermartingale processes introduced by…
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to…
We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and for stationary sequences the…
This work provides a novel convergence analysis for stochastic optimization in terms of stopping times, addressing the practical reality that algorithms are often terminated adaptively based on observed progress. Unlike prior approaches,…
We give sufficient conditions for the bounded law of the iterated logarithms for strictly stationary random fields when the summation is done on rectangle. The study is done by the control of an appropriated maximal function. The case of…
This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic…
Consider a discrete-time martingale, and let $V^2$ be its normalized quadratic variation. As $V^2$ approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian…
We develop a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities…
Viewing a two time scale stochastic approximation scheme as a noisy discretization of a singularly perturbed differential equation, we obtain a concentration bound for its iterates that captures its behavior with quantifiable high…
We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the…
The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and…
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using the Taylor expansion, is…
In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is…
When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient…
The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, general central limit theorems and functional…
The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the…
For martingales with a wide range of integrability, we will quantify the rate of convergence of the central limit theorem via Wasserstein distances of order $r$, $1\le r\le 3$. Our bounds are in terms of Lyapunov's coefficients and the…
A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal chaining-based methods. The proposed approach begins by deriving…
We generalise the martingale-coboundary representation of discrete time stochastic processes to the non-stationary case and to random variables in Orlicz spaces. Related limit theorems (CLT, invariance principle, log log law, probabilities…
We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured…