Related papers: An equivalent conjecture to Feige's Conjecture

We study an information analogue of infinitely divisible probability distributions, where the i.i.d. sum is replaced by the joint distribution of an i.i.d. sequence. A random variable $X$ is called informationally infinitely divisible if,…

The problem of combining p-values is an old and fundamental one, and the classic assumption of independence is often violated or unverifiable in many applications. There are many well-known rules that can combine a set of arbitrarily…

We obtain some new results concerning the small deviation problem for $S=\sum_n q^n X_n$ and $M=\sup_n q^n X_n$, where $0<q<1$ and $(X_n)$ are i.i.d. non-negative random variables. In particular, the asymptotics is shown to be the same for…

An analogue of the Berry-Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n^{-1/2}, the same as in the classical case.…

Let $A_n$ be a random symmetric matrix with Bernoulli $\{\pm 1\}$ entries. For any $\kappa>0$ and two real numbers $\lambda_1,\lambda_2$ with a separation $|\lambda_1-\lambda_2|\geq \kappa n^{1/2}$ and both lying in the bulk…

One-parameter natural exponential family (NEF) plays fundamental roles in probability and statistics. This article contains two independent results: (a) A conjecture of Bar-Lev, Bshouty and Enis states that a polynomial with a simple root…

Let $f(n)$ denote a multiplicative function with range $\{-1,0,1\}$, and let $F(x) = \sum_{n\leq x} f(n)$. Then $F(x)/\sqrt{x} = a\sqrt{x} + b + E(x)$, where $a$ and $b$ are constants and $E(x)$ is an error term that either tends to $0$ in…

We investigate the complete $p$-th moment convergence for weighted sums of independent, identically distributed random variables under sublinear expectations space. Using moment inequality and truncation methods, we prove the equivalent…

A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: For each pair of positive integers $(n,k)$ with $3\le k\le n-3$, there is a $k$-dimensional $\Bbb F_2$-subspace $E$…

This paper is devoted to the study of $\Phi$-moments of sums of independent/freely independent random variables. More precisely, let $(f_k)_{k=1}^n$ be a sequence of positive (symmetrically distributed) independent random variables and let…

Let $A_{\pi}(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $\pi$ for $\rm SL_3(\mathbb{Z})$ and $ \omega(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum…

For a given testing problem, let $U_1,...,U_n$ be individually valid and conditionally on the data i.i.d.\ P-variables (often called P-values). For example, the data could come in groups, and each $U_i$ could be based on subsampling just…

The complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space was studied. By moment inequality and truncation methods, we establish the equivalent…

We determine the probability $P$ of two independent events $A$ and $B$, which occur randomly $n_A$ and $n_B$ times during a total time $T$ and last for $t_A$ and $t_B$, to occur simultaneously at some point during $T$. Therefore we first…

We propose a safe approximation to joint chance-constrained programming where the constraint functions are additively dependent on a normally-distributed random vector. The approximation is analytical, meaning that it requires neither…

I proposed (8, 1, 3) that p values should be supplemented by an estimate of the false positive risk (FPR). FPR was defined as the probability that, if you claim that there is a real effect on the basis of p value from a single unbiased…

Sums of of 1-dependent integer-valued random variables are approximated by compound Poisson, negative binomial and Binomial distributions and signed compound Poisson measures. Estimates are obtained for total variation and local metrics.…

We give a new bound on the probability that the random sum $\xi_1 v_1 +...+ \xi_n v_n$ belongs to a ball of fixed radius, where the $\xi_i$ are iid Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an application, we prove…

Let $X$ be an irreducible shift of finite type (SFT) of positive entropy, and let $B_n(X)$ be its set of words of length $n$. Define a random subset $\omega$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some…

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 =…