Given two sequences over a finite alphabet $\mathcal{L}$, the D₂ statistic is the number of m-letter word matches between the two sequences. This statistic is used in bioinformatics for expressed sequence tag database searches. Here we study a generalization of the D₂ statistic in the context of DNA sequences, under the assumption of strand symmetric Bernoulli text. For k<m, we look at the count of m-letter word matches with up to k mismatches. For this statistic, we compute the expectation, give upper and lower bounds for the variance and prove its distribution is asymptotically normal.

This work considers a server that processes J classes using the generalized processor sharing discipline with base weight vector α=(α₁, …, α_J) and redistribution weight vector β=(β₁, …, β_J). The invariant manifold $\mathcal{M}$ of the so-called fluid limit associated with this model is shown to have the form $\mathcal{M}=\{x\in\mathbb{R}_{+}^{J}:x_{j}=0\mbox{ for }j\in\mathcal{S}\}$, where $\mathcal{S}$ is the set of strictly subcritical classes, which is identified explicitly in terms of the vectors α and β and the long-run average work arrival rates γ_j of each class j. In addition, under general assumptions, it is shown that when the heavy traffic condition ∑_j=1^Jγ_j=∑_j=1^Jα_j holds, the functional central limit of the scaled unfinished work process is a reflected diffusion process that lies in $\mathcal{M}$. The reflected diffusion limit is characterized by the so-called extended Skorokhod map and may fail to be a semimartingale. This generalizes earlier results obtained for the simpler, balanced case where γ_j=α_j for j=1, …, J, in which case $\mathcal{M}=\mathbb{R}_{+}^{J}$ and there is no state-space collapse. Standard techniques for obtaining diffusion approximations cannot be applied in the unbalanced case due to the particular structure of the GPS model. Along the way, this work also establishes a comparison principle for solutions to the extended Skorokhod map associated with this model, which may be of independent interest.

This paper studies variations of the usual voter model that favor types that are locally less common. Such models are dual to certain systems of branching annihilating random walks that are parity preserving. For both the voter models and their dual branching annihilating systems we determine all homogeneous invariant laws, and we study convergence to these laws started from other initial laws.

When gas molecules bind to a surface they may do so in such a way that the adsorption of one molecule inhibits the arrival of others. We consider random sequential adsorption in which the empty sites of a graph are irreversibly occupied in random order by a variety of types of “particles.” In a finite region the process terminates when no more particles can arrive. A universal asymptotic formula for the mean duration is given.

We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is well known that without disorder there are no interface Gibbs measures in infinite volume in dimension d=2, while there are “gradient Gibbs measures” describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn.

In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in d=2. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation.

In d=3 where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.

Let (X_t)_t≥0 be a continuous-time, time-homogeneous strong Markov process with possible jumps and let τ be its first hitting time of a Borel subset of the state space. Suppose X is sampled at random times and suppose also that X has not hit the Borel set by time t. What is the intensity process of τ based on this information?

This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin–Yor [Lecture Notes in Math. 649 (1978) 78–97] result regarding compensators under a general filtration expansion framework, a novel computation methodology for the intensity process of a stopping time is proposed. En route, an analogous characterization result for martingales of Jacod and Skorohod [Lecture Notes in Math. 1583 (1994) 21–35] under local jumping filtration is derived.

In this paper we lay the foundation for a numerical algorithm to simulate high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular, we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the number of iterations. This feature seems to be indispensable for an efficient iterative scheme from a numerical point of view. We finally suggest a fully explicit numerical algorithm and present some numerical examples with up to 10-dimensional state space.

We consider the problem of the error in soliton transmission in long-haul optical fibers caused by the spontaneous emission of noise inherent to amplification. We study two types of noises driving the stochastic focusing cubic one dimensional nonlinear Schrödinger equation which appears in physics in that context. We focus on the fluctuations of the mass and arrival time or timing jitter. We give the small noise asymptotic of the tails of these two quantities for the two types of noises. We are then able to prove several results from physics among which the Gordon–Haus effect which states that the fluctuation of the arrival time is a much more limiting factor than the fluctuation of the mass. The physical results had been obtained with arguments difficult to fully justify mathematically.

We consider branching random walks built on Galton–Watson trees with offspring distribution having a bounded support, conditioned to have n nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of “globally centered discrete snake” that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when n goes to +∞, “globally centered discrete snakes” converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton–Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node u is the vector indexed by (k, j) giving the number of ancestors of u having k children and for which u is a descendant of the jth one]. Some consequences concerning Galton–Watson trees conditioned by the size are also derived.

Under the continuous assumption on the generator g, Briand et al. [Electron. Comm. Probab. 5 (2000) 101–117] showed some connections between g and the conditional g-expectation $({\mathcal{E}}_{g}[\cdot|{\mathcal{F}}_{t}])_{t\in[0,T]}$ and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19–34] showed some connections between g and the corresponding dynamic risk measure (ρ_t^g)_{t∈[0, T]}. In this paper we prove that, without the additional continuous assumption on g, a g-expectation ${\mathcal{E}}_{g}$ satisfies translation invariance if and only if g is independent of y, and ${\mathcal{E}}_{g}$ satisfies convexity (resp. subadditivity) if and only if g is independent of y and g is convex (resp. subadditive) with respect to z. By these conclusions we deduce that the static risk measure ρ^g induced by a g-expectation ${\mathcal{E}}_{g}$ is a convex (resp. coherent) risk measure if and only if g is independent of y and g is convex (resp. sublinear) with respect to z. Our results extend the results in Briand et al. [Electron. Comm. Probab. 5 (2000) 101–117] and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19–34] on these subjects.

An evolutionarily stable strategy (ESS) is an equilibrium strategy that is immune to invasions by rare alternative (“mutant”) strategies. Unlike Nash equilibria, ESS do not always exist in finite games. In this paper we address the question of what happens when the size of the game increases: does an ESS exist for “almost every large” game? Letting the entries in the n×n game matrix be independently randomly chosen according to a distribution F, we study the number of ESS with support of size 2. In particular, we show that, as n→∞, the probability of having such an ESS: (i) converges to 1 for distributions F with “exponential and faster decreasing tails” (e.g., uniform, normal, exponential); and (ii) converges to $1-1/\sqrt{e}$ for distributions F with “slower than exponential decreasing tails” (e.g., lognormal, Pareto, Cauchy).

Our results also imply that the expected number of vertices of the convex hull of n random points in the plane converges to infinity for the distributions in (i), and to 4 for the distributions in (ii).

We study the profile X_n,k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for k=⌊αlogn⌋ in a certain range of α.

A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.

Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability ν, or a voter model on ℤ in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if Mν/L and L/K² converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.