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Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2007)

ESAIM: Probability and Statistics

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x → ∞, where T x ˜ denotes a suitable renormalization of Tx.


Asymptotic behaviour of stochastic systems with conditionally exponential decay property

Agnieszka Jurlewicz, Aleksander Weron, Karina Weron (1996)

Applicationes Mathematicae

A new class of CED systems, providing insight into behaviour of physical disordered materials, is introduced. It includes systems in which the conditionally exponential decay property can be attached to each entity. A limit theorem for the normalized minimum of a CED system is proved. Employing different stable schemes the universal characteristics of the behaviour of such systems are derived.

Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks

Stefan Adams, Tony Dorlas (2008)

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

We study large deviations principles for N random processes on the lattice ℤd with finite time horizon [0, β] under a symmetrised measure where all initial and terminal points are uniformly averaged over random permutations. That is, given a permutation σ of N elements and a vector (x1, …, xN) of N initial points we let the random processes terminate in the points (xσ(1), …, xσ(N)) and then sum over all possible permutations and initial points, weighted with an initial distribution. We prove level-two...

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Olivier Garet, Régine Marchand (2004)

ESAIM: Probability and Statistics

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on ℤ d to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical...

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Olivier Garet, Régine Marchand (2010)

ESAIM: Probability and Statistics

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on ℤ d to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical...

Asymptotics of counts of small components in random structures and models of coagulation-fragmentation

Boris L. Granovsky (2013)

ESAIM: Probability and Statistics

We establish necessary and sufficient conditions for the convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. The multiplicative measures depict distributions of component spectra of random structures and also the equilibria of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We show that the convergence of multiplicative measures is equivalent to the asymptotic independence of counts of...

Averaged large deviations for random walk in a random environment

Atilla Yilmaz (2010)

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

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤd with d≥1, and gives a variational formula for the corresponding rate function Ia. Under Sznitman’s transience condition (T), we show that Ia is strictly convex and analytic on a non-empty open set , and that the true velocity of the particle is an element (resp. in the boundary) of when the walk is non-nestling...

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