We reveal a shape transition for a transient simple random walk forced to realize an excess q-norm of the local times, as the parameter q crosses the value qc(d)=d/(d−2). Also, as an application of our approach, we establish a central limit theorem for the q-norm of the local times in dimension 4 or more.

The life distribution H(t) of a device subject to shocks governed by a Poisson process and pure birth process is considered as a function of probabilities Pk of not surviving the first k shocks. It is shown that some properties of a discrete distribution {P'k} are reflected on properties of the continuous life distribution H(t). In particular, if Pk has the discrete NBUFR properties, then H(t) has the continuous NBUFR and NBAFR properties. The NBUFR and NBAFR life distributions are obtained under...

Markov chain usage models were successfully used to model systems and software. The most prominent approaches are the so-called failure state models Whittaker and Thomason (1994) and the arc-based Bayesian models Sayre and Poore (2000). In this paper we propose arc-based semi-Markov usage models to test systems. We extend previous studies that rely on the Markov chain assumption to the more general semi-Markovian setting. Among the obtained results we give a closed form representation of the first...

Les processus de Schramm-Loewner (SLE) induisent des courbes aléatoires du plan complexe, qui vérifient une propriété d’invariance conforme. Ce sont des outils fondamentaux pour la compréhension du comportement asymptotique en régime critique de certains modèles discrets intervenant en physique statistique ; ils ont permis notamment d’établir rigoureusement certaines conjectures importantes dans ce domaine.

This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on ${\mathbb{Z}}^d$, when $d>2$. We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important role in...

We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition $\left(T^{\text{'}}\right)$. We show that for every $ϵ>0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $exp(-{(logn)}^{d-ϵ})$. This bound almost matches the known lower bound of $exp(-C{(logn)}^d)$, and significantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched probability...

This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate thestabilitiesof 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...

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 the exponential...

For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1<α<2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time...