We consider the stepping stone model on the torus of side L in ℤ² in the limit L→∞, and study the time it takes two lineages tracing backward in time to coalesce. Our work fills a gap between the finite range migration case of [Ann. Appl. Probab. 15 (2005) 671–699] and the long range case of [Genetics 172 (2006) 701–708], where the migration range is a positive fraction of L. We obtain limit theorems for the intermediate case, and verify a conjecture in [Probability Models for DNA Sequence Evolution (2008) Springer] that the model is homogeneously mixing if and only if the migration range is of larger order than (log L)^1/2.

A d-dimensional binary Markov random field on a lattice torus is considered. As the size n of the lattice tends to infinity, potentials a=a(n) and b=b(n) depend on n. Precise bounds for the probability for local configurations to occur in a large ball are given. Under some conditions bearing on a(n) and b(n), the distance between copies of different local configurations is estimated according to their weights. Finally, a sufficient condition ensuring that a given local configuration occurs everywhere in the lattice is suggested.

There is a growing interest in the literature for adaptive Markov chain Monte Carlo methods based on sequences of random transition kernels {P_n} where the kernel P_n is allowed to have an invariant distribution π_n not necessarily equal to the distribution of interest π (target distribution). These algorithms are designed such that as n→∞, P_n converges to P, a kernel that has the correct invariant distribution π. Typically, P is a kernel with good convergence properties, but one that cannot be directly implemented. It is then expected that the algorithm will inherit the good convergence properties of P. The equi-energy sampler of [Ann. Statist. 34 (2006) 1581–1619] is an example of this type of adaptive MCMC. We show in this paper that the asymptotic variance of this type of adaptive MCMC is always at least as large as the asymptotic variance of the Markov chain with transition kernel P. We also show by simulation that the difference can be substantial.

We consider a random walk in an i.i.d. Cauchy-tailed conductances environment. We obtain a quenched functional CLT for the suitably rescaled random walk, and, as a key step in the arguments, we improve the local limit theorem for p_n²t^ω(0, y) in [Ann. Probab. (2009). To appear], Theorem 5.14, to a result which gives uniform convergence for p_n²t^ω(x, y) for all x, y in a ball.

Suppose that X₁, …, X_n, … are i.i.d. rotationally invariant N-by-N matrices. Let Π_n=X_n⋯X₁. It is known that n⁻¹log |Π_n| converges to a nonrandom limit. We prove that under certain additional assumptions on matrices X_i the speed of convergence to this limit does not decrease when the size of matrices, N, grows.

In this paper we prove a functional limit theorem for the weighted profile of a b-ary tree. For the proof we use classical martingales connected to branching Markov processes and a generalized version of the profile-polynomial martingale. By embedding, choosing weights and a branch factor in a right way, we finally rediscover the profiles of some well-known discrete time trees.

We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence and for the presence of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains.

We consider the accumulation of beneficial and deleterious mutations in large asexual populations. The rate of adaptation is affected by the total mutation rate, proportion of beneficial mutations and population size N. We show that regardless of mutation rates, as long as the proportion of beneficial mutations is strictly positive, the adaptation rate is at least $\mathcal{O}(\log^{1-\delta}N)$ where δ can be any small positive number, if the population size is sufficiently large. This shows that if the genome is modeled as continuous, there is no limit to natural selection, that is, the rate of adaptation grows in N without bound.

Ewens sampling formula (ESF) is a one-parameter family of probability distributions with a number of intriguing combinatorial connections. This elegant closed-form formula first arose in biology as the stationary probability distribution of a sample configuration at one locus under the infinite-alleles model of mutation. Since its discovery in the early 1970s, the ESF has been used in various biological applications, and has sparked several interesting mathematical generalizations. In the population genetics community, extending the underlying random-mating model to include recombination has received much attention in the past, but no general closed-form sampling formula is currently known even for the simplest extension, that is, a model with two loci. In this paper, we show that it is possible to obtain useful closed-form results in the case the population-scaled recombination rate ρ is large but not necessarily infinite. Specifically, we consider an asymptotic expansion of the two-locus sampling formula in inverse powers of ρ and obtain closed-form expressions for the first few terms in the expansion. Our asymptotic sampling formula applies to arbitrary sample sizes and configurations.

We propose an unconstrained stochastic approximation method for finding the optimal change of measure (in an a priori parametric family) to reduce the variance of a Monte Carlo simulation. We consider different parametric families based on the Girsanov theorem and the Esscher transform (exponential-tilting). In [Monte Carlo Methods Appl. 10 (2004) 1–24], it described a projected Robbins–Monro procedure to select the parameter minimizing the variance in a multidimensional Gaussian framework. In our approach, the parameter (scalar or process) is selected by a classical Robbins–Monro procedure without projection or truncation. To obtain this unconstrained algorithm, we extensively use the regularity of the density of the law without assuming smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions as well as for diffusion processes.

We illustrate the efficiency of our algorithm on several pricing problems: a Basket payoff under a multidimensional NIG distribution and a barrier options in different markets.

Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only through experimental observation or numerical simulations. In both cases, the data on which the topological measurements are based are derived via some form of finite sampling or discretization. In this paper, we present a probabilistic approach to quantifying the number of components of generalized nodal domains of nonhomogeneous random processes on the real line via finite discretizations, that is, we consider excursion sets of a random process relative to a nonconstant deterministic threshold function. Our results furnish explicit probabilistic a priori bounds for the suitability of certain discretization sizes and also provide information for the choice of location of the sampling points in order to minimize the error probability. We illustrate our results for a variety of random processes, demonstrate how they can be used to sample the classical nodal domains of deterministic functions perturbed by additive noise and discuss their relation to the density of zeros.

The antique Mills Futurity slot machine has two unusual features. First, if a player loses 10 times in a row, the 10 lost coins are returned. Second, the payout distribution varies from coup to coup in a manner that is nonrandom and periodic with period 10. It follows that the machine is driven by a 100-state irreducible period-10 Markov chain. Here, we evaluate the stationary distribution of the Markov chain, and this leads to a strong law of large numbers and a central limit theorem for the sequence of payouts. Following a suggestion of Pyke [In Mathematical Statistics and Applications: Festschrift for Constance van Eeden (2003) 185–216 Institute of Mathematical Statistics], we address the question of whether there exists a two-armed version of this “one-armed bandit” that obeys Parrondo’s paradox. More precisely, is there such a machine with the property that the casino can honestly advertise that both arms are fair, yet when players alternate arms in certain random or nonrandom ways, the casino makes money in the long run? The answer is a qualified yes. Although this “history-dependent” game is conceptually simpler than the original such games of Parrondo, Harmer and Abbott [Phys. Rev. Lett. (2000) 85 5226–5229], it is nearly as complicated analytically, and open problems remain.

There has recently been considerable interest in the stability of different fair bandwidth sharing policies for models that arise in the context of Internet congestion control. Here, we consider a connection level model, introduced by Massoulié and Roberts [Telecommunication Systems 15 (2000) 185–201], that represents the randomly varying number of flows present in a network. The weighted α-fair and weighted max–min fair bandwidth sharing policies are among important policies that have been studied for this model. Stability results are known in both cases when the interarrival times and service times are exponentially distributed. Partial results for general service times are known for weighted α-fair policies; no such results are known for weighted max–min fair policies. Here, we show that weighted max–min fair policies are stable for subcritical networks with general interarrival and service distributions, provided the latter have 2+δ₁ moments for some δ₁>0. Our argument employs an appropriate Lyapunov function for the weighted max–min fair policy.