There is a one-to-one correspondence between classical one-dimensional infinitely divisible distributions and free infinitely divisible distributions. In this work we study the free infinitely divisible distributions corresponding to the one-dimensional type G distributions. A new characterization of classical type G distributions is given first and the class of type A classical infinitely divisible distributions is introduced. The corresponding free type A distributions are studied and the role of a special symmetric beta distribution is shown as a building block for free type A distributions. It is proved that this symmetric beta distribution is the free multiplicative convolution of an arcsine distribution with the Marchenko–Pastur distribution.

Convergence in Mallows distance is of particular interest when heavy-tailed distributions are considered. For 1≤α<2, it constitutes an alternative technique to derive Central Limit type theorems for non-Gaussian α-stable laws. In this note, for properly stabilized martingale sums and sequences of ϕ-mixing random variables, we establish Mallows convergence to stable laws. Sufficient conditions are presented in the setting of familiar Lindeberg-like conditions and extend earlier results for the independent case.

Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems.

Consider a random walk {X_n : n≥0} in an elliptic i.i.d. environment in dimensions d≥2 and call P₀ its averaged law starting from 0. Given a direction $l\in\mathbb{S}^{d-1}$, A_l={lim_n→∞ X_n⋅l=∞} is called the event that the random walk is transient in the direction l. Recently Simenhaus proved that the following are equivalent: the random walk is transient in the neighborhood of a given direction; P₀-a.s. there exists a deterministic asymptotic direction; the random walk is transient in any direction contained in the open half space defined by this asymptotic direction. Here we prove that the following are equivalent: P₀(A_l∪A_−l)=1 in the neighborhood of a given direction; there exists an asymptotic direction ν such that P₀(A_ν∪A_−ν)=1 and P₀-a.s we have $\lim_{n\to\infty}X_{n}/|X_{n}|=𝟙_{A_{\nu}}\nu-𝟙_{A_{-\nu}}\nu$; P₀(A_l∪A_−l)=1 if and only if l⋅ν≠0. Furthermore, we give a review of some open problems.

We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar rotor models during stochastic time evolution are presented.

We consider a two-type (red and blue or R and B) particle population that evolves on the d-dimensional lattice according to some reaction-diffusion process R+B→2R and starts with a single red particle and a density ρ of blue particles. For two classes of models we give an upper bound on the propagation velocity of the red particles front with explicit dependence on ρ.

In the first class of models red particles evolve with a diffusion constant D_R=1. Blue particles evolve with a possibly time-dependent jump rate D_B≥0, or, more generally, follow independent copies of some bistochastic process. Examples of bistochastic process also include long-range random walks with drift and various deterministic processes. For this class of models we get in all dimensions an upper bound of order $\max(\rho,\sqrt{\rho})$ that depends only on ρ and d and not on the specific process followed by blue particles, in particular that does not depend on D_B. We argue that for d≥2 or ρ≥1 this bound can be optimal (in ρ), while for the simplest case with d=1 and ρ<1 known as the frog model, we give a better bound of order ρ.

In the second class of models particles evolve according to Kawasaki dynamics, that is, with exclusion and possibly attraction, inside a large two-dimensional box with periodic boundary conditions (this turns into simple exclusion when the attraction is set to zero). In a low density regime we then get an upper bound of order $\sqrt{\rho}$. This proves a long-range decorrelation of dynamical events in this low density regime.

We describe some recent results concerning the statistical properties of a self-interacting polymer stretched by an external force. We concentrate mainly on the cases of purely attractive or purely repulsive self-interactions, but our results are stable under suitable small perturbations of these pure cases. We provide in particular a precise description of the stretched phase (local limit theorems for the endpoint and local observables, invariance principle, microscopic structure). Our results also characterize precisely the (nontrivial, direction-dependent) critical force needed to trigger the collapsed/stretched phase transition in the attractive case. We also describe some recent progress: first, the determination of the order of the phase transition in the attractive case; second, a proof that a semi-directed polymer in quenched random environment is diffusive in dimensions 4 and higher when the temperature is high enough. In addition, we correct an incomplete argument from Ioffe and Velenik [In Analysis and Stochastics of Growth Processes and Interface Models (2008) 55–79].

Let {X(v), v∈ℤ^d×ℤ₊} be an i.i.d. family of random variables such that P{X(v)=e^b}=1−P{X(v)=1}=p for some b>0. We consider paths π⊂ℤ^d×ℤ₊ starting at the origin and with the last coordinate increasing along the path, and of length n. Define for such paths W(π)= number of vertices π_i, 1≤i≤n, with X(π_i)=e^b. Finally, let N_n(α)= number of paths π of length n starting at π₀=0 and with W(π)≥αn. We establish several properties of lim_n→∞[N_n]^1/n.

We find upper bounds for the probability of underestimation and overestimation errors in penalized likelihood context tree estimation. The bounds are explicit and applies to processes of not necessarily finite memory. We allow for general penalizing terms and we give conditions over the maximal depth of the estimated trees in order to get strongly consistent estimates. This generalizes previous results obtained in the case of estimation of the order of a Markov chain.

We extend the strong macroscopic stability introduced in Bramson and Mountford [Ann. Probab. 30 (2002) 1082–1130] for one-dimensional asymmetric exclusion processes with finite range to a large class of one-dimensional conservative attractive models (including misanthrope process) for which we relax the requirement of finite range kernels. A key motivation is the extension of constructive hydrodynamics result of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1–30, Ann. Probab. 34 (2006) 1339–1369, Electron. J. Probab. (to appear)] to nonfinite range kernels.

This paper gives a didactic introduction to the problem of metastability by taking elementary examples from the theory of random walks. Some mathematical tools of the theory are presented briefly with precise references. The main ideas used in recent results about the conservative case, are discussed in the last section, through simplified models of random walks.

We present results of Monte Carlo simulation and chaos approximation of a class of Markov processes with a countable or continuous set of states. Each of these states can be written as a finite (finite case) or infinite in both directions (infinite case) sequence of pluses and minuses denoted by ⊕ and ⊖. As continuous time goes on, our sequence undergoes the following three types of local transformations: the first one, called flip, changes any minus into plus and any plus into minus with a rate β; the second, called annihilation, eliminates two neighbor components with a rate α whenever they are in differents states; and the third, called mitosis, doubles any component with a rate γ. All of them occur at any place of the sequence independently. Our simulations and approximations suggest that with appropriate positive values of α, β and γ this process has the following two properties. Growth: In the finite case, as the process goes on, the length of the sequence tends to infinity with a probability which tends to 1 when the length of the initial sequence tends to ∞. Nonergodicity: The infinite process is nonergodic and the finite process keeps most of the time at two extremes, occasionally swinging from one to the other.

The paper proposes a class of financial market models which are based on inhomogeneous telegraph processes and jump diffusions with alternating volatilities. It is assumed that the jumps occur when the tendencies and volatilities are switching. Such a model captures well the stock price dynamics under periodic financial cycles. The distribution of this process is described in detail. We also provide a closed form of the structure of risk-neutral measures. This incomplete model can be completed by adding another asset based on the same sources of randomness. For completed market model we obtain explicit formulae for call prices.