Related papers: A note on the binomial distribution motivated by C…

An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We prove a structural…

The present article derives the minimal number $N$ of observations needed to consider a Bayesian posterior distribution as Gaussian. Two examples are presented. Within one of them, a chi-squared distribution, the observable $x$ as well as…

Let $O(2n+\ell)$ be the group of orthogonal matrices of size $\left(2n+\ell\right)\times \left(2n+\ell\right)$ equipped with the probability distribution given by normalized Haar measure. We study the probability \begin{equation*}…

Let $n$ be a sufficiently large natural number and let $B$ be an origin-symmetric convex body in $R^n$ in the $\ell$-position, and such that the normed space $(R^n,\|\cdot\|_B)$ admits a $1$-unconditional basis. Then for any…

We prove that the empirical spectral distribution of a (d_L, d_R)-biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Mar\v{c}enko-Pastur distribution of random matrix theory. This convergence…

For a graph $G$ and $p\in[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p\geq C(\log n)^{1/3}n^{-2/3}$…

We prove that the distribution density of any non-constant polynomial $f(\xi_1,\xi_2,\ldots)$ of degree $d$ in independent standard Gaussian random variables $\xi$ (possibly, in infinitely many variables) always belongs to the…

In this work, we propose a simplification of the Pollaczek-Khinchine formula for the ultimate time survival (or ruin) probability calculation in exchange for a few assumptions on the random variables which generate the renewal risk model.…

The bipartite Ramsey number $B(n_1,n_2,\ldots,n_t)$ is the least positive integer $b$ such that, any coloring of the edges of $K_{b,b}$ with $t$ colors will result in a monochromatic copy of $K_{n_i,n_i}$ in the $i-$th color, for some $i$,…

Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the…

Every graph $G=(V,E)$ is an induced subgraph of some Kneser graph of rank $k$, i.e., there is an assignment of (distinct) $k$-sets $v \mapsto A_v$ to the vertices $v\in V$ such that $A_u$ and $A_v$ are disjoint if and only if $uv\in E$. The…

Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros…

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high…

Let $\varphi_{n,K}$ denote the largest angle in all the triangles with vertices among the $n$ points selected at random in a compact convex subset $K$ of $\mathbb{R}^d$ with nonempty interior, where $d\ge2$. It is shown that the…

In this short note we show that a system $M=(E,r)$ with a ground set $E$ of size $m$ and (rank) function $r: 2^E\to \mathbb{Z}_{\geq 0}$ satisfying $r(S)\leq \min(r(E),|S|)$ for every set $S\subseteq E$, the Tutte polynomial…

The article studies the almost surely asymptotics of extreme values $\bar{\xi}_n = \max_{1\leq i \leq n} \xi_i$, where $ \xi , \xi_1 , \xi_2 , \ldots$ are discrete identically distributed random variables. One of the main results on this…

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a…

Let $ X_1, \ldots, X_n $ be independent random variables taking values in the alphabet $ \{0, 1, \ldots, r\} $, and $ S_n = \sum_{i = 1}^n X_i $. The Shepp--Olkin theorem states that, in the binary case ($ r = 1 $), the Shannon entropy of $…

We consider the problem of finding anomalies in a $d$-dimensional field of independent random variables $\{Y_i\}_{i \in \left\{1,...,n\right\}^d}$, each distributed according to a one-dimensional natural exponential family $\mathcal F =…

Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any nonnegative random vector. Theorem 1.2 requires multivariate size bias…