Small positive values for supercritical branching processes in random environment

Vincent Bansaye; Christian Böinghoff

Displaying similar documents to “Small positive values for supercritical branching processes in random environment”

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.

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

Frank Aurzada, Steffen Dereich (2013)

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

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

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...

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...

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...

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...

Limit distributions for multitype branching processes of $m$ -ary search trees

Brigitte Chauvin, Quansheng Liu, Nicolas Pouyanne (2014)

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

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Let $m \geq 3$ 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 $m \leq 26$ , the asymptotic behavior of the process is Gaussian, but for $m \geq 27$ it is no longer Gaussian and a limit $W^{D T}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

Lévy processes conditioned on having a large height process

Mathieu Richard (2013)

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

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In the present work, we consider spectrally positive Lévy processes $(X_{t}, t \geq 0)$ not drifting to $+ \infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$ ) before hitting $0$ . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $ℙ_{x}^{☆}$ of this conditioned process (starting at $x g t; 0$ ) is defined as a Doob $h$ -transform via a martingale. For Lévy processes with infinite variation paths,...

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...

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.

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...

On the strong Brillinger-mixing property of $α$ -determinantal point processes and some applications

Lothar Heinrich (2016)

Applications of Mathematics

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First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C (x, y)$ defining an $α$ -determinantal point process (DPP). Assuming absolute integrability of the function $C_{0} (x) = C (o, x)$ , we show that a stationary $α$ -DPP with kernel function $C_{0} (x)$ is “strongly” Brillinger-mixing, implying, among others, that its tail- $σ$ -field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...

The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert, Florian Simatos (2014)

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

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We consider regenerative processes with values in some general Polish space. We define their $ε$ -big excursions as excursions $e$ such that $ϕ (e) g t; ε$ , where $ϕ$ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of $e$ . We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of $ε$ -big excursions and of their endpoints, for all $ε$ in a set whose closure contains $0$ . Finally,...

Weak convergence of mutually independent $X ₙ^{B}$ and $X ₙ^{A}$ under weak convergence of $X ₙ \equiv X ₙ^{B} - X ₙ^{A}$

W. Szczotka (2006)

Applicationes Mathematicae

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For each n ≥ 1, let $v_{n, k}, k \geq 1$ and $u_{n, k}, k \geq 1$ 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 $X ₙ^{B} (t) = (1 / c ₙ) \sum_{j = 1}^{[n t]} (v_{n, j} - v ̅ ₙ)$ , $X ₙ^{A} (t) = (1 / c ₙ) \sum_{j = 1}^{[n t]} (u_{n, j} - u ̅ ̅ ₙ)$ , t ≥ 0, and $X ₙ = X ₙ^{B} - X ₙ^{A}$ . The main result gives conditions under which the weak convergence $X ₙ \overset{}{\to} X$ , where X is a Lévy process, implies $X ₙ^{B} \overset{}{\to} X^{B}$ and $X ₙ^{A} \overset{}{\to} X^{A}$ , where $X^{B}$ and $X^{A}$ are mutually independent Lévy processes and $X = X^{B} - X^{A}$ .

Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

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A notion of a wide-sense Markov process $X_{t}$ of order k ≥ 1, $X_{t} \sim W M (k)$ , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of $X_{t}$ is the k-dimensional process $x_{t} = (X_{t - k + 1}, . . ., X_{t})$ . The covariance structure of $X_{t} \sim W M (k)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_{t} \sim W M (k)$ iff $x_{t}$ is a k-dimensional WM(1) process and iff the covariance function of $x_{t}$ has the triangular...

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.

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...

On smoothing properties of transition semigroups associated to a class of SDEs with jumps

Seiichiro Kusuoka, Carlo Marinelli (2014)

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

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We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in $ℝ^{d}$ driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y = W \circ T$ , where $T$ is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process $W$ ) and of an arbitrary Lévy process independent of $Y$ , that the drift coefficient is continuous...