We consider a branching-selection system in ℝ with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c ≥ a or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener–Hopf equations.

Random walks in random scenery are processes defined by Z_n := ∑_k=1ⁿ ξ_X1+⋯+Xk, where (X_k, k ≥ 1) and (ξ_y, y ∈ ℤ) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2], respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α ≠ 1 and as n → ∞, of n^−δ Z_n, for some suitable δ > 0 depending on α and β. Here, we are interested in the convergence, as n → ∞, of n^δℙ(Z_n = ⌊n^δ x⌋), when x ∈ ℝ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

We study a generalized Pólya urn model with two types of ball. If the drawn ball is red, it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colors are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colors swap, the process is recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting, our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth–death processes, a uniform renewal process, the Eulerian numbers, and Lamperti’s problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally, we discuss some related models of independent interest, including a “Poisson earthquakes” Markov chain on the homeomorphisms of the plane.

We generalize a result from Volkov [Ann. Probab. 29 (2001) 66–91] and prove that, on a large class of locally finite connected graphs of bounded degree (G, ∼) and symmetric reinforcement matrices a = (a_i,j)_i,j∈G, the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete d-partite subgraph with possible loops plus its outer boundary.

We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph G satisfies the property that its support is a complete d-partite subgraph of G with possible loops, for some d ≥ 1. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory.

Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117–136] and Benaïm [Ann. Probab. 25 (1997) 361–392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support S, the following event occurs with positive probability: the walk localizes on S ∪ ∂ S (where ∂ S is the outer boundary of S) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood.

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension β* + 1, where β* = (√17 − 3)/2, and that it can be described as the geodesic lamination coded by a random continuous function which is Hölder continuous with exponent β* − ε, for every ε > 0. We also discuss recursive constructions of triangulations of the n-gon that give rise to the same continuous limit when n tends to infinity.

Let Π_n be the set of convex polygonal lines Γ with vertices on ℤ₊² and fixed endpoints 0 = (0, 0) and n = (n₁, n₂). We are concerned with the limit shape, as n → ∞, of “typical” Γ ∈ Π_n with respect to a parametric family of probability measures {P_n^r, 0 < r < ∞} on Π_n, including the uniform distribution (r = 1) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. Bárány and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class {P_n^r}, even though P_n^r (r ≠ 1) and P_n¹ are asymptotically singular. Measures P_n^r are constructed, following Sinai’s approach, as conditional distributions Q_z^r(⋅|Π_n), where Q_z^r are suitable product measures on the space Π = ⋃_n Π_n, depending on an auxiliary “free” parameter z = (z₁, z₂). The transition from (Π, Q_z^r) to (Π_n, P_n^r) is based on the asymptotics of the probability Q_z^r(Π_n), furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the Möbius inversion formula and properties of zeroes of the Riemann zeta function.

In this paper, we develop a general machinery for finding explicit uniform probability and moment bounds on sub-additive positive functionals of random processes. Using the developed general technique, we derive uniform bounds on the ${\mathbb{L}}_{s}$-norms of empirical and regression-type processes. Usefulness of the obtained results is illustrated by application to the processes appearing in kernel density estimation and in nonparametric estimation of regression functions.

Given a stable Lévy process X = (X_t)_0≤t≤T of index α ∈ (1, 2) with no negative jumps, and letting S_t = sup_0≤s≤t  X_s denote its running supremum for t ∈ [0, T], we consider the optimal prediction problem $$V = \inf_{0≤τ≤T}\mathsf E(S_T − X_τ)^p, $$ where the infimum is taken over all stopping times τ of X, and the error parameter p ∈ (1, α) is given and fixed. Reducing the optimal prediction problem to a fractional free-boundary problem of Riemann–Liouville type, and finding an explicit solution to the latter, we show that there exists α_∗ ∈ (1, 2) (equal to 1.57 approximately) and a strictly increasing function p_∗ : (α_∗, 2) → (1, 2) satisfying p_∗(α_∗+) = 1, p_∗(2−) = 2 and p_∗(α) < α for α ∈ (α_∗, 2) such that for every α ∈ (α_∗, 2) and p ∈ (1, p_∗(α)) the following stopping time is optimal $$τ_∗ = \inf\{t ∈ [0, T] : S_t − X_t ≥ z_∗(T − t)^{1/α}\},$$ where z_∗ ∈ (0, ∞) is the unique root to a transcendental equation (with parameters α and p). Moreover, if either α ∈ (1, α_∗) or p ∈ (p_∗(α), α) then it is not optimal to stop at t ∈ [0, T) when S_t − X_t is sufficiently large. The existence of the breakdown points α_∗ and p_∗(α) stands in sharp contrast with the Brownian motion case (formally corresponding to α = 2), and the phenomenon itself may be attributed to the interplay between the jump structure (admitting a transition from lighter to heavier tails) and the individual preferences (represented by the error parameter p).

We give an elementary proof of the celebrated Bichteler–Dellacherie theorem which states that the class of stochastic processes S allowing for a useful integration theory consists precisely of those processes which can be written in the form S = M + A, where M is a local martingale and A is a finite variation process. In other words, S is a good integrator if and only if it is a semi-martingale.

We obtain this decomposition rather directly from an elementary discrete-time Doob–Meyer decomposition. By passing to convex combinations, we obtain a direct construction of the continuous time decomposition, which then yields the desired decomposition.

As a by-product of our proof, we obtain a characterization of semi-martingales in terms of a variant of no free lunch, thus extending a result from [Math. Ann. 300 (1994) 463–520].

The results of Komlós, Major and Tusnády give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [Ann. Probab. 35 (2007) 2294–2320] obtained Wiener approximation of a class of dependent stationary processes with finite pth moments, 2 < p ≤ 4, with error term o(n^1/p(log n)^γ), γ > 0, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249–280] removed the logarithmic factor, reaching the Komlós–Major–Tusnády bound o(n^1/p). No similar results exist for p > 4, and in fact, no existing method for dependent approximation yields an a.s. rate better than o(n^1/4). In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to o(n^1/p) for arbitrary p > 2. This extends the scope of applications of the results essentially, as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term (epidemic) changes in stationary processes. Our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems.

We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type measures. We find conditions for such a measure to have a smooth density and give examples. The Hunt semigroup and generator of convolution semigroups of measures are represented as pseudo-differential operators. For sufficiently regular convolution semigroups, the transition kernel has a tractable Fourier expansion and the density at the neutral element may be expressed as the trace of the Hunt semigroup. We compute the short time asymptotics of the density at the neutral element for the Cauchy distribution on the d-torus, on SU(2) and on SO(3), where we find markedly different behaviour than is the case for the usual heat kernel.