Related papers: Some notes on concentration for $\alpha$-subexpone…
(English) This monograph aims at presenting the core weak convergence theory for sequences of random vectors with values in $\mathbb{R}^k$. In some places, a more general formulation in metric spaces is provided. It lays out the necessary…
We show that any probability measure satisfying a Matrix Poincar\'e inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carr\'e du champ…
Convolutions of long-tailed and subexponential distributions play a major role in the analysis of many stochastic systems. We study these convolutions, proving some important new results through a simple and coherent approach, and showing…
In this paper, we generalize and improve some fundamental concentration inequalities using information on the random variables' higher moments. In particular, we improve the classical Hoeffding's and Bennett's inequalities for the case…
We consider the large deviations associated with the empirical mean of independent and identically distributed random variables under a subexponential moment condition. We show that non-trivial deviations are observable at a subexponential…
We obtain Rosenthal-type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremisers in log-concave settings when the moments of summands are…
For a wide class of monotonic functions $f$, we develop a Chernoff-style concentration inequality for quadratic forms $Q_f \sim \sum\limits_{i=1}^n f(\eta_i) (Z_i + \delta_i)^2$, where $Z_i \sim N(0,1)$. The inequality is expressed in terms…
Consider a random sum $\eta_1 v_1 + ... + \eta_n v_n$, where $\eta_1,...,\eta_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $\P(\eta_1 v_1 + ...…
A lower bound on the probability $P(0<X<\delta)$ for all real $\delta>0$ and all random variables $X$ with log-concave p.d.f.'s such that $EX=0$ and $EX^2=1$ is obtained.
We consider random walks with finite second moment which drifts to $-\infty$ and have heavy tail. We focus on the events when the minimum and the final value of this walk belong to some compact set. We first specify the associated…
The right tail asymptotic series consisting of attenuating exponential terms are derived for the densities of Galton-Watson processes with fractional probability generating functions. The frequencies in the exponential factors form fractal…
The "typical" asymptotic behavior of the weighted sums of independent, identically distibuted random vectors in k-dimensional space is considered. It is shown that under finitnes of fifth absolute moment of an individual term the rate of…
We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding…
Recently, the complete left tail asymptotic for the density of the {\it martingale limit} of the classical Galton-Watson process has been derived. The derivation is based on the properties of a special function (whose inverse Fourier…
We consider the sequence of independent centered Gaussian random elements of a separable Banach space and their consecutive closed convex hulls. If inicial elements converge weakly to some limite, then, as shown in Davydov- Paulauskas…
Concentration inequalities for the sample mean, like those due to Bernstein, Hoeffding, and Bentkus, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem…
In this paper we define the closure under weak convergence of the class of p-tempered {\alpha}-stable distributions. We give necessary and sufficient conditions for convergence of sequences in this class. Moreover, we show that any element…
The matrix Markov inequality by Ahlswede was stated using the Loewner anti-order between positive definite matrices. Wang use this to derive several other Chebyshev and Chernoff-type inequalities (Hoeffding, Bernstein, empirical Bernstein)…
Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In…
We propose new concentration inequalities for self-normalized martingales. The main idea is to introduce a suitable weighted sum of the predictable quadratic variation and the total quadratic variation of the martingale. It offers much more…