Related papers: A note on a new exponential bound for M-acceptable…

Let $\lambda$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{\alpha \in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} \lambda(n) e(n\alpha)\bigg\rvert = o(HX)$$ as $X\to \infty$, in the regime $H =…

In this article, the complete moment convergence for the partial sum of moving average processes $\{X_n=\sum_{i=-\infty}^{\infty}a_iY_{i+n},n\ge 1\}$ is estabished under some proper conditions, where $\{Y_i,-\infty<i<\infty\}$ is a sequence…

The following problem is considered: given a joint distribution $P_{XY}$ and an event $E$, bound $P_{XY}(E)$ in terms of $P_XP_Y(E)$ (where $P_XP_Y$ is the product of the marginals of $P_{XY}$) and a measure of dependence of $X$ and $Y$.…

We establish explicit exponential convergence estimates for the renewal theorem, in terms of a uniform component of the inter arrival distribution, of its Laplace transform which is assumed finite on a positive interval, and of the Laplace…

The average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value $z$, random variables $X_1, \ldots, X_n$, and an error parameter $\varepsilon > 0$, and we…

The entropy power inequality (EPI) provides lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have…

In this text we study, for positive random variables, the relation between the behaviour of the Laplace transform near infinity and the distribution near zero. A result of De Bruijn shows that $E(e^{-\lambda X}) \sim \exp(r\lambda^\alpha)$…

It is established that for any finite set of positive real numbers $A$, we have $$|A/A+A| \gg \frac{|A|^{\frac{3}{2}+\frac{1}{26}}}{\log^{1/2}|A|}.$$

The small sample universal hypothesis testing problem is investigated in this paper, in which the number of samples $n$ is smaller than the number of possible outcomes $m$. The goal of this work is to find an appropriate criterion to…

Writing for a general mathematical audience, we provide elementary upper and lower bounds on the growth (as a function of N) of the sum \sum_{n=1}^N (-1)^{\floor{n x}} for various fixed x. For example, if x is a quadratic irrational, then…

The idea of the restricted mean has been used to establish a significantly improved version of Markov's inequality that does not require any new assumptions. The result immediately extends on Chebyshev's inequalities and Chernoff's bound.…

We extend a general Bernstein-type maximal inequality of Kevei and Mason (2011) for sums of random variables.

We show that the $L^1$ norm of an exponential sum of length $X$ and with coefficients equal to the Liouville or M\"{o}bius function is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville…

The paper provides new upper and lower bounds for the multivariate Laplace approximation under weak local assumptions. Their range of validity is also given. An application to an integral arising in the extension of the Dixon's identity is…

A random phenomenon may have two sources of random variation: an unstable identity and a set of external variation-generating factors. When only a single source is active, two mutually exclusive extreme scenarios may ensue that result in…

We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which…

Young's integral inequality is complemented with an upper bound to the remainder. The new inequality turns out to be equivalent to Young's inequality, and the cases in which the equality holds become particularly transparent in the new…

The Bruss-Robertson inequality gives a bound on the maximal number of elements of a random sample whose sum is less than a specified value, and the extension of that inequality which is given here neither requires the independence of the…

For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied…

Let $X$ be a random variable with finite second moment. We investigate the inequality: $P\{|X-E[X]|\le \sqrt{{\rm Var}(X)}\}\ge P\{|Z|\le 1\}$, where $Z$ is a standard normal random variable. We prove that this inequality holds for many…