Universality of the asymptotics of the one-sided exit problem for integrated processes

Frank Aurzada; Steffen Dereich

Displaying similar documents to “Universality of the asymptotics of the one-sided exit problem for integrated processes”

The spread of a catalytic branching random walk

Philippe Carmona, Yueyun Hu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a catalytic branching random walk on $ℤ$ 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 $α$ , $\frac{M_{n}}{n} \to α$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_{n} - α n$ as $n$ goes to infinity.

Positivity of integrated random walks

Vladislav Vysotsky (2014)

Annales de l'I.H.P. Probabilités et statistiques

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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 $α$ -stable law with $1 l t; α \leq 2$ . Assuming that $S_{1}$ is either right-exponential (i.e. $ℙ (S_{1} g t; x | S_{1} g t; 0) = e^{- a x}$ for some $a g t; 0$ and all $x g t; 0$ ) or right-continuous (skip free), we prove that $ℙ {A_{1} g t; 0, \dots, A_{N} g t; 0} \sim C_{α} N^{1 / (2 α) - 1 / 2}$ as $N \to \infty$ , where $C_{α} g t; 0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Giant component and vacant set for random walk on a discrete torus

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

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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 $log N$ . Moreover, this connected component occupies a...

Random ε-nets and embeddings in $ℓ_{\infty}^{N}$

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

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

About the generating function of a left bounded integer-valued random variable

Charles Delorme, Jean-Marc Rinkel (2008)

Bulletin de la Société Mathématique de France

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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 $1 - Φ (z)$ inside the unit disk, where $Φ$ is the generating function of $X$ , under some mild conditions

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

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We study a continuous time growth process on the $d$ -dimensional hypercubic lattice $𝒵^{d}$ , which admits a phenomenological interpretation as the combustion reaction $A + B \to 2 A$ , 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...

Small positive values for supercritical branching processes in random environment

Vincent Bansaye, Christian Böinghoff (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Branching Processes in Random Environment (BPREs) $(Z_{n} : n \geq 0)$ 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 $ℙ (1 \leq Z_{n} \leq k | Z_{0} = i)$ , $k, i \in ℕ$ as $n \to \infty$ . More precisely, we characterize...

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let Ω be a countable infinite product $Ω ₁^{ℕ}$ 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 $E_{*} [X ₁] \leq l i m i n f S ₙ / n \leq l i m s u p S ₙ / n \leq E * [X ₁]$ , where $E_{*}$ and E* are the lower and upper expectations....

The critical barrier for the survival of branching random walk with absorption

Bruno Jaffuel (2012)

Annales de l'I.H.P. Probabilités et statistiques

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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...

Collisions of random walks

Martin T. Barlow, Yuval Peres, Perla Sousi (2012)

Annales de l'I.H.P. Probabilités et statistiques

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A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$ , started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton–Watson tree with finite variance conditioned to survive, the incipient infinite cluster in $ℤ^{d}$ with $d \geq 19$ and the uniform spanning tree in $ℤ^{2}$ all have the infinite collision property. For power-law combs and spherically...

Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin, Ross A. Maller (2013)

Annales de l'I.H.P. Probabilités et statistiques

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This paper is concerned with the small time behaviour of a Lévy process $X$ . In particular, we investigate theof the times, ${\bar{T}}_{b} (r)$ and $T_{b}^{*} (r)$ , at which $X$ , started with $X_{0} = 0$ , first leaves the space-time regions ${(t, y) \in ℝ^{2} : y \leq r t^{b}, t \geq 0}$ (one-sided exit), or ${(t, y) \in ℝ^{2} : | y | \leq r t^{b}, t \geq 0}$ (two-sided exit), $0 \leq b l t; 1$ , as $r ↓ 0$ . 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...

Random walks on co-compact fuchsian groups

Sébastien Gouëzel, Steven P. Lalley (2013)

Annales scientifiques de l'École Normale Supérieure

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It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence $R$ . It is also shown that Ancona’s inequalities extend to $R$ , and therefore that the Martin boundary for $R$ -potentials coincides with the natural geometric boundary $S^{1}$ , and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic...

Stable random fields and geometry

Shigeo Takenaka (2010)

Banach Center Publications

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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 $Y (B); B \in ℬ (S^{m})$ 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,..., $B_{i} \cap B_{j} = \emptyset$ ,...

The absolute continuity of the invariant measure of random iterated function systems with overlaps

Balázs Bárány, Tomas Persson (2010)

Fundamenta Mathematicae

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We consider iterated function systems on the interval with random perturbation. Let $Y_{ε}$ be uniformly distributed in [1-ε,1+ ε] and let $f_{i} \in C^{1 + α}$ be contractions with fixpoints $a_{i}$ . We consider the iterated function system $Y_{ε} f_{i} + a_{i} (1 - Y_{ε}) ⁿ_{i = 1}$ , 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...

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky, Yongfeng Wu (2015)

Applications of Mathematics

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

Large scale behaviour of the spatial $𝛬$ -Fleming–Viot process

N. Berestycki, A. M. Etheridge, A. Véber (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the spatial $𝛬$ -Fleming–Viot process model ( (2010) 162–216) for frequencies of genetic types in a population living in $ℝ^{d}$ , in the special case in which there are just two types of individuals, labelled $0$ and $1$ . At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$ . We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the...

Limit theorems for geometric functionals of Gibbs point processes

T. Schreiber, J. E. Yukich (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Observations are made on a point process $𝛯$ in $ℝ^{d}$ in a window $Q_{λ}$ of volume $λ$ . The observation, or ‘score’ at a point $x$ , here denoted $ξ (x, 𝛯)$ , is a function of the points within a random distance of $x$ . When the input $𝛯$ is a Poisson or binomial point process, the large $λ$ limit theory for the total score $\sum_{x \in 𝛯 \cap Q_{λ}} ξ (x, 𝛯 \cap Q_{λ})$ , when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

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In this paper we establish a decoupling feature of the random interlacement process $ℐ^{u} \subset ℤ^{d}$ at level $u$ , $d \geq 3$ . Roughly speaking, we show that observations of $ℐ^{u}$ restricted to two disjoint subsets $A_{1}$ and $A_{2}$ of $ℤ^{d}$ 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...