We show that, given an n-dimensional normed space X, a sequence of $N={(8/\epsilon )}^{2n}$ independent random vectors ${\left(X_i\right)}_{i=1}^N$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $\Gamma :ℝ\to {ℝ}^N$ defined by $\Gamma x={(⟨x,X_i⟩)}_{i=1}^N$ embeds X in ${\ell }_{\infty }^N$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into ${\ell }_{\infty }^N$ with asymptotically best possible relation between N, n, and ε.

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1-\Phi \left(z\right)$ inside the unit disk, where $\Phi$ is the generating function of $X$, under some mild conditions

Let $\{X_{n,j},1\le j\le m\left(n\right),n\ge 1\}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0<b_n\to \infty$. Conditions are given for ${\sum }_{j=1}^{m\left(n\right)}{X}_{n,j}/b_n\to 0$ completely and for ${max}_{1\le k\le m\left(n\right)}\left|{\sum }_{j=1}^k{X}_{n,j}\right|/b_n\to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums ${\sum }_{j=1}^{m\left(n\right)}{X}_{n,j}^{*}$ of the dependent bootstrap samples $\{\{X_{n,j}^{*},1\le j\le m\left(n\right)\},n\ge 1\}$ arising from a sequence of i.i.d. random variables $\{X_n,n\ge 1\}$.

Take a centered random walk $S_n$ and consider the sequence of its partial sums $A_n:={\sum }_{i=1}^n{S}_i$. Suppose $S_1$ is in the domain of normal attraction of an $\alpha$-stable law with $1lt;\alpha \le 2$. Assuming that $S_1$ is either right-exponential (i.e. $\mathbb{P}(S_{1}gt;x|S_{1}gt;0)={\mathrm{e}}^{-ax}$ for some $agt;0$ and all $xgt;0$) or right-continuous (skip free), we prove that $$\mathbb{P}\{A_{1}gt;0,\cdots ,A_{N}gt;0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$ as $N\to \infty$, where $C_{\alpha }gt;0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

We study a continuous time growth process on the $d$-dimensional hypercubic lattice ${𝒵}^d$, which admits a phenomenological interpretation as the combustion reaction $A+B\to 2A$, where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...

Let 1 ≤ p < 2 and let $L_p=L_p[0,1]$ be the classical $L_p$-space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable $f\in L_p$ spans in $L_p$ a subspace isomorphic to some Orlicz sequence space $l_M$. We give precise connections between M and f and establish conditions under which the distribution of a random variable $f\in L_p$ whose independent copies span $l_M$ in $L_p$ is essentially unique.

In this paper we consider a random variable $h$ arising from an intersection problem between a fixed convex body $K_0$ and a system of random independent and uniformly distributed ovaloids in $E_3$.

The slicing problem can be reduced to the study of isotropic convex bodies K with $diam\left(K\right)\le c\surd n{L}_K$, where $L_K$ is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that ${\left|\right|⟨·,\theta ⟩\left|\right|}_{\psi ₂}\le C{L}_K$ for all θ in a subset U of $S^{n-1}$ with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that $ma{x}_{\theta \in S^{n-1}}{\left|\right|⟨·,\theta ⟩\left|\right|}_{\psi ₂}\ge c∜n{L}_K$. In a different direction, we show that good average ψ₂-behaviour of linear functionals...

We consider iterated function systems on the interval with random perturbation. Let $Y_{\epsilon }$ be uniformly distributed in [1-ε,1+ ε] and let $f_i\in C^{1+\alpha }$ be contractions with fixpoints $a_i$. We consider the iterated function system $Y_{\epsilon }{f}_i+a_i(1-Y_{\epsilon }){ⁿ}_{i=1}$, where each of the maps is chosen with probability $p_i$. It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection...

We consider random walk on a discrete torus $E$ of side-length $N$, in sufficiently high dimension $d$. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u{N}^d$. We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $logN$. Moreover, this connected component occupies a...

Let Ω be a countable infinite product $\Omega {₁}^{\mathbb{N}}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*}\left[X₁\right]\le liminfSₙ/n\le limsupSₙ/n\le E*\left[X₁\right]$, where $E_{*}$ and E* are the lower and upper expectations....

A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable $X_p$ such that $X\stackrel{d}{=}{\sum }_{k=1}^{T\left(p\right)}{X}_{p,k}$, where $X_{p,k}$’s are i.i.d. copies of $X_p$, and random variable T(p) independent of $X_{p,1},X_{p,2},...$ has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The main results concern geometrically strictly semistable distributions which form a subset of geometrically infinitely divisible distributions....

Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for $M=S^m$, the m-dimensional sphere. Let $Y\left(B\right);B\in ℬ\left(S^m\right)$ be the Gaussian random measure on $S^m$, that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on $S^m$, 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for $B_i$, i = 1,2,..., $B_i\cap B_j=\varnothing$,...

Consider a $N\times n$ non-centered matrix ${𝛴}_n$ with a separable variance profile: $${𝛴}_n=\frac{D_n^{1/2}{X}_n{\tilde{D}}_n^{1/2}}{\sqrt{n}}+A_n.$$ Matrices $D_n$ and ${\tilde{D}}_n$ are non-negative deterministic diagonal, while matrix $A_n$ is deterministic, and $X_n$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_n\left(z\right)$ the resolvent associated to ${𝛴}_n{𝛴}_n^{*}$, i.e. $$Q_n\left(z\right)={\left({𝛴}_n{𝛴}_n^{*}-z{I}_N\right)}^{-1}.$$ Given two sequences of deterministic vectors $\left(u_n\right)$ and $\left(v_n\right)$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:...

We consider a catalytic branching random walk on $\mathbb{Z}$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$: For some constant $\alpha$, $\frac{M_n}{n}\to \alpha$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_n-\alpha n$ as $n$ goes to infinity.

The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{\nu }\left(E₁\times E₂\right)=min(C_{\nu ₁}\left(E₁\right),C_{\nu ₂}\left(E₂\right))$, where $E_j$ and ${\nu }_j$ are respectively a compact set and a norm in ${ℂ}^{N_j}$ (j = 1,2), and ν is a norm in ${ℂ}^{N₁+N₂}$, ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of ${ℂ}^N$, denote by C(E) the standard L-capacity and by ${\omega }_E$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

Let ${ₙ}^{\left(k\right)}$ be a family of random independent k-element subsets of [n] = 1,2,...,n and let $({ₙ}^{\left(k\right)},\ell )={ₙ}^{\left(k\right)}\left(\ell \right)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to ${ₙ}^{\left(k\right)}\left(\ell \right)$ depends only on the edges of ${ₙ}^{\left(k\right)}$ contained in S. Then, the edges of ${ₙ}^{\left(k\right)}\left(\ell \right)$ are ’weakly dependent’, say, the events that two given subsets S and T are in ${ₙ}^{\left(k\right)}\left(\ell \right)$ are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We...

In this paper we establish a decoupling feature of the random interlacement process ${ℐ}^u\subset {\mathbb{Z}}^d$ at level $u$, $d\ge 3$. Roughly speaking, we show that observations of ${ℐ}^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of ${\mathbb{Z}}^d$ are approximately independent, once we add a sprinkling to the process ${ℐ}^u$ by slightly increasing the parameter $u$. Our results differ from previous ones in that we allow the mutual distance between the sets $A_1$ and $A_2$ to be much smaller than their diameters. We then provide an important application...

We improve the constants in the Men’shov-Rademacher inequality by showing that for n ≥ 64, $E\left(su{p}_{1\le k\le n}\right|{\sum }_{i=1}^k{X}_i|²\le 0.11(6.20+log₂n)²$ for all orthogonal random variables X₁,..., Xₙ such that ${\sum }_{k=1}^{n}E\left|X_k\right|²=1$.

In this paper, we study the size of the giant component $C_G$ in the random geometric graph $G=G(n,r_n,f)$ of $n$ nodes independently distributed each according to a certain density $f(·)$ in ${[0,1]}^2$ satisfying ${inf}_{x\in {[0,1]}^2}f\left(x\right)gt;$ $0$. If $\frac{c_1}{n}\le r_n^2\le c_2\frac{logn}{n}$ for some positive constants $c_1$, $c_2$ and $n{r}_n^2⟶\infty$ as $n\to \infty$, we show that the giant component of $G$ contains at least $n-\mathrm{o}\left(n\right)$ nodes with probability at least $1-{\mathrm{e}}^{-\beta n{r}_n^2}$ for all $n$ and for some positive constant $\beta$. We also obtain estimates on the diameter and number of the non-giant components of $G$.

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space ${ℝ}^n$. It is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $GL\left(n\right)$ covariant, and associative if and only if it is $L_p$ addition for some $1\le p\le \infty$. It is also demonstrated...

We introduce Oberwolfach randomness, a notion within Demuth’s framework of statistical tests with moving components; here the components’ movement has to be coherent across levels. We show that a ML-random set computes all $K$-trivial sets if and only if it is not Oberwolfach random, and indeed that there is a $K$-trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis,...

Let $S_n^{\left(2\right)}$ denote the iterated partial sums. That is, $S_n^{\left(2\right)}=S_1+S_2+\cdots +S_n$, where $S_i=X_1+X_2+\cdots +X_i$. Assuming $X_1,X_2,...,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}lt;0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$ with $c\le 6\sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{\left(2\right)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0lt;\gamma lt;1/4$ there exist integrable,...

Suppose that $𝒢$ is a finite, connected graph and $X$ is a lazy random walk on $𝒢$. The lamplighter chain $X^{\diamond }$ associated with $X$ is the random walk on the wreath product ${𝒢}^{\diamond }={𝐙}_2\wr 𝒢$, the graph whose vertices consist of pairs $(\underline{f},x)$ where $f$ is a labeling of the vertices of $𝒢$ by elements of ${𝐙}_2=\{0,1\}$ and $x$ is a vertex in $𝒢$. There is an edge between $(\underline{f},x)$ and $(\underline{g},y)$ in ${𝒢}^{\diamond }$ if and only if $x$ is adjacent to $y$ in $𝒢$ and $f_z=g_z$ for all $z\ne x,y$. In each step, $X^{\diamond }$ moves from a configuration $(\underline{f},x)$ by updating $x$ to $y$ using the transition rule of $X$ and then...