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...

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 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...

This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate theof the times, ${\overline{T}}_b\left(r\right)$ and $T_b^{*}\left(r\right)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (one-sided exit), or $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (two-sided exit), $0\le blt;1$, as $r\downarrow 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^p$. In many instances these are...

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.

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

The authors first establish the Marcinkiewicz-Zygmund inequalities with exponent $p$ ($1\le p\le 2$) for $m$-pairwise negatively quadrant dependent ($m$-PNQD) random variables. By means of the inequalities, the authors obtain some limit theorems for arrays of rowwise $m$-PNQD random variables, which extend and improve the corresponding results in [Y. Meng and Z. Lin (2009)] and [H. S. Sung (2013)]. It is worthy to point out that the open problem of [H. S. Sung, S. Lisawadi, and A. Volodin (2008)] can be...

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...

Let $m\ge 3$ be an integer. The so-calledis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m\le 26$, the asymptotic behavior of the process is Gaussian, but for $m\ge 27$ it is no longer Gaussian and a limit $W^{DT}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

Branching Processes in Random Environment (BPREs) $(Z_n:n\ge 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $\mathbb{P}(1\le Z_n\le k|Z_0=i)$, $k,i\in \mathbb{N}$ as $n\to \infty$. More precisely, we characterize...

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 study a branching random walk on $ℝ$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. [ (1991) 573–581] determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a...

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 $G$.

We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha$-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta \left(\alpha \right)$ such that the probability that any $\alpha$-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta \left(\alpha \right)+\mathrm{o}\left(1\right)}$ for large $T$. We also investigate when the fixed level can be replaced by a different...

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,...

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.

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:...