In this paper we give new deviation inequalities for the partial sums of weakly dependent data. The loss from the independent case is studied carefully. We give examples of non mixing time series such that dynamical systems and Bernoulli shifts…

We consider a branching random walk on $R$ with a killing barrier at zero. At criticality, the process becomes eventually extinct, and the total progeny $Z$ is therefore finite. We show that $P(Z>n)$ is of order  $(n\ln^2(n))^{-1}$, which confirms the…

We give new exponential inequalities for the spectral measure of random Wishart matrices. These results give in particular useful bounds when these matrices have the form $M=YY^T$, in the case where $Y$ is a $p\times n$ random matrix with independent…

We study the critical parameter $u^*$ of random interlacements on a Galton-Watson tree conditioned on the non-extinction event. We show that, for a given law of a Galton-Watson tree, the value of this parameter is a.s. constant and non-trivial. We…

For two-dimensional last-passage time models of weakly increasing paths, interesting scaling limits have been proved for points close the axis (the hard edge). For strictly increasing paths of Bernoulli($p$) marked sites, the relevant boundary is the line $y=px$. We call…

We prove the existence of a (random) Lipschitz function $F:\mathbb{Z}^{d-1}\to\mathbb{Z}^+$ such that, for every $x\in\mathbb{Z}^{d-1}$, the site $(x,F(x))$ is open in a site percolation process on $\mathbb{Z}^{d}$. The Lipschitz constant may be taken to be $1$ when the parameter $p$…

For $0<\alpha\leq 2$ and $0<\beta\leq 1$ let $X$ be the $(d,\alpha,\beta)$-superprocess, i.e. the superprocess with $\alpha$-stable spatial movement in $R^d$ and $(1+\beta)$-stable branching. Given that the initial measure $X_0$ is Lebesgue on $R^d$, Iscoe conjectured in [7] that the total…

We describe the scaling limit of the nearest neighbour prudent walk on $Z^2$, which performs steps uniformly in directions in which it does not see sites already visited. We show that the scaling limit is given by the process $Z_u…

We explore the principle of smooth fit in the case of the discounted optimal stopping problem $$ V(x)=\sup_\tau\, \mathsf{E}_x[e^{-\beta\tau}G(X_\tau)]. $$ We show that there exists a regular diffusion $X$ and differentiable gain function $G$ such that the value function $V$…

Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in \mathbb{T}}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\Lambda(\cdot)$. Let $m_n$ denote the…