Related papers: Freedman's inequality for matrix martingales
Consider $n$ complex random matrices $X_1,\ldots,X_n$ of size $d\times d$ sampled i.i.d. from a distribution with mean $E[X]=\mu$. While the concentration of averages of these matrices is well-studied, the concentration of other functions…
We derive a new maximal inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe [Ann. Probab. 28 (2000) 713-724]. Then, we apply it to establish the Donsker invariance principle for this…
We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size $N$ converge to the Tracy--Widom laws at a rate $O(N^{-1/3+\omega})$, as $N$ tends to infinity. For Wigner matrices this…
We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
In this paper we study the path-regularity and martingale properties of the set-valued stochastic integrals defined in our previous work Ararat et al. (2023). Such integrals have some fundamental differences from the well-known…
In this article the relation between the tail behaviours of a free regular infinitely divisible (positively supported) probability measure and its L\'evy measure is studied. An important example of such a measure is the compound free…
We present an accelerated, or 'look-ahead' version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current…
We establish Hoeffding-type concentration inequalities for the low and high tail bounds of sums of exchangeable random variables. Our results exhibit an anti-symmetry in such tail bounds due to the assumption of exchangeability, a…
The fractional Leibniz rule is generalized by the Coifman-Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.
We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call \emph{permutation mixtures}) and their i.i.d. counterparts. Our results are based on a novel method for controlling $\chi^2$…
Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et…
In this note we prove the following good-$\lambda$ inequality, for $r>2$, all $\lambda > 0$, $\delta \in \big(0, \frac{1}{2} \big)$ \[ \nu\big\{ V_r(f) > 3 \lambda ; \mathcal{M}(f) \leq \delta \lambda\big\} \leq 4 \nu\{s(f) > \delta…
We give a distribution-dependent concentration inequality for functions of independent variables. The result extends Bernstein's inequality from sums to more general functions, whose variation in any argument does not depend too much on the…
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 discrete Schr\"odinger equation with the Dirichlet boundary condition is considered on a half-line lattice when the potential is real valued and compactly supported. The inverse problem of recovery of the potential from the so-called…
We provide a systematic approach to stable central limit theorems for d-dimensional martingale difference arrays and martingale difference sequences. The conditions imposed are straightforward extensions of the univariate case.
Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in…
We give an alternative proof of a sharp generalization of an integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables, is possible. This last…
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…
We study deterministic and stochastic primal-dual sub-gradient algorithms for distributed optimization of a separable objective function with global inequality constraints. In both algorithms, the norm of the Lagrangian multipliers are…