arXiv subjects

Dainius Dzindzalieta

Publications and source records attributed to Dainius Dzindzalieta.

5 recordsLinked to original sources

arXiv2019

A non-uniform Littlewood-Offord inequality

Consider a sum $S_n=v_i\varepsilon_1+\cdots+v_n\varepsilon_{n}$, where $(v_i)^{n}_{i=1}$ are non-zero vectors in $\mathbb{R}^{d}$ and $(\varepsilon_i)^{n}_{i=1}$ are independent Rademacher random variables (i.e., $~{\mathbb{P}(\varepsilon_{i}=\pm 1)=1/2}$). The classical Littlewood-Offord problem asks for the best possible upper bound for $~{\sup_{x}\mathbb{P}(S_n = x)}$. In this paper we consider a non-uniform version of this problem. Namely, we obtain the optimal bound for $\mathbb{P}(S_n = x)$ in terms of the length of the vector $x\in \mathbb{R}^d$.

math.PR↗

arXiv2015

A tight Gaussian bound for weighted sums of Rademacher random variables

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent identically distributed Rademacher random variables, that is $\mathbb{P}\{\varepsilon_i=\pm1\}=1/2$. Let $S_n=a_1\varepsilon_1+\cdots+a_n\varepsilon_n$, where $\mathbf{a}=(a_1,\ldots,a_n)\in\mathbb{R}^n$ is a vector such that ${a_1^2+\cdots+a_n^2\leq1}$. We find the smallest possible constant $c$ in the inequality \[\mathbb{P}\{S_n\geq x\}\leq c\mathbb{P}\{η\geq x\}\qquad for all x\in \mathbb{R},\] where $η\sim N(0,1)$ is a standard normal random variable. This optimal value is equal to \[c_*=\bigl(4\mathbb{P}\{η\geq\sqrt{2}\}\bigr)^ {-1}\approx3.178.\]

math.PR↗

arXiv2013

Random walks maximizing the probability to visit an interval

We consider random walks, say $W_n=(M_0, M_1,\dots, M_n)$, of length $n$ starting at 0 and based on the martingale sequence $M_k$ with differences $X_m=M_m-M_{m-1}$. Assuming that the differences are bounded, $|X_m|\leq 1$, we solve the problem \begin{equation} D_n(x)\=\sup P \left\{W_n \ \text{visits an interval}\ [x,\infty)\right\},\qquad x\in R, \label{piirma} \end{equation} where $\sup$ is taken over all possible $W_n$. In particular, we describe random walks which maximize the probability in $\eqref{piirma}$. We also extend the result to super-martingales.

math.PR↗

arXiv2013

Extremal Lipschitz functions in the deviation inequalities from the mean

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}μ\{f-\E_μ f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of $\eqref{abstr}$ for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\R^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove that in general probability metric spaces the $\sup$ in $\eqref{abstr}$ is achieved on a family of distance functions.

math.PR↗

arXiv2012

Optimal Probability Inequalities for Random Walks related to Problems in Extremal Combinatorics

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=ε_{1}+...+ε_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that \mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}, where A is either an interval of the form [x, \infty) or just a single point. The inequality is exact and the optimal values of c and k are given explicitly. It improves Kwapień's inequality in the case of the Rademacher series. We also provide a new and very short proof of the Littlewood-Offord problem without using Sperner's Theorem. Finally, an extension to odd Lipschitz functions is given.

math.PR↗