We consider a spatial branching process with emigration in which children either remain at the same site as their parents or migrate to new locations and then found their own colonies. We are interested in asymptotics of the partition of the total population into colonies for large populations with rare migrations. Under appropriate regimes, we establish weak convergence of the rescaled partition to some random measure that is constructed from the restriction of a Poisson point measure to a certain random region, and whose cumulant solves a simple integral equation.

For any finite colored graph we define the empirical neighborhood measure, which counts the number of vertices of a given color connected to a given number of vertices of each color, and the empirical pair measure, which counts the number of edges connecting each pair of colors. For a class of models of sparse colored random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distribution of Erdős–Rényi graphs near criticality.

The one-dimensional motion of any number ${\mathcal{N}}$ of particles in the field of many independent waves (with strong spatial correlation) is formulated as a second-order system of stochastic differential equations, driven by two Wiener processes. In the limit of vanishing particle mass ${\mathfrak{m}}\to0$, or, equivalently, of large noise intensity, we show that the momenta of all ${\mathcal{N}}$ particles converge weakly to ${\mathcal{N}}$ independent Brownian motions, and this convergence holds even if the noise is periodic. This justifies the usual application of the diffusion equation to a family of particles in a unique stochastic force field. The proof rests on the ergodic properties of the relative velocity of two particles in the scaling limit.

We analyze the asymptotic properties of a Euclidean optimization problem on the plane. Specifically, we consider a network with three bins and n objects spatially uniformly distributed, each object being allocated to a bin at a cost depending on its position. Two allocations are considered: the allocation minimizing the bin loads and the allocation allocating each object to its less costly bin. We analyze the asymptotic properties of these allocations as the number of objects grows to infinity. Using the symmetries of the problem, we derive a law of large numbers, a central limit theorem and a large deviation principle for both loads with explicit expressions. In particular, we prove that the two allocations satisfy the same law of large numbers, but they do not have the same asymptotic fluctuations and rate functions.

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that:

 (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices;

 (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval almost surely;

 (iii) the empirical distributions of the eigenvalues of the Laplacian matrices converge weakly to the free convolution of the standard Gaussian distribution and the Wigner’s semi-circular law;

 (iv) the empirical distributions of the eigenvalues of the adjacency matrices converge weakly to the Wigner’s semi-circular law.

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume–Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter α governing the speed at which the sequence approaches criticality is below a certain threshold α₀. However, when α exceeds α₀, the thermodynamic magnetization converges to 0 much faster than the finite-size magnetization. The asymptotic behavior of the finite-size magnetization is proved via a moderate deviation principle when 0 < α < α₀ and via a weak-convergence limit when α > α₀. To the best of our knowledge, our results are the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model.

We consider the problem of maximizing expected utility from terminal wealth in models with stochastic factors. Using martingale methods and a conditioning argument, we determine the optimal strategy for power utility under the assumption that the increments of the asset price are independent conditionally on the factor process.

This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen [Bernoulli 7 (2001) 223–242] for target distributions with a noncompact support. The conditions ensuring a strong law of large numbers require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab. 16 (2006) 1462–1505].

This work considers a many-server queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measure-valued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essentially only requiring that the service and reneging distributions have densities, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a fluid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.

The present paper studies the limiting behavior of the average score of a sequentially selected group of items or individuals, the underlying distribution of which, F, belongs to the Gumbel domain of attraction of extreme value distributions. This class contains the Normal, Lognormal, Gamma, Weibull and many other distributions. The selection rules are the “better than average” (β = 1) and the “β-better than average” rule, defined as follows. After the first item is selected, another item is admitted into the group if and only if its score is greater than β times the average score of those already selected. Denote by Y̅_k the average of the k first selected items, and by T_k the time it takes to amass them. Some of the key results obtained are: under mild conditions, for the better than average rule, Y̅_k less a suitable chosen function of logk converges almost surely to a finite random variable. When 1 − F(x) = e^{−[xα + h(x)]}, α > 0 and h(x) / x^α →^{x → ∞} 0, then T_k is of approximate order k². When β > 1, the asymptotic results for Y̅_k are of a completely different order of magnitude. Interestingly, for a class of distributions, T_k, suitably normalized, asymptotically approaches 1, almost surely for relatively small β ≥ 1, in probability for moderate sized β and in distribution when β is large.

We describe all countable particle systems on ℝ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure $\mathfrak{m}$ and moving independently of each other according to the law of some Gaussian process ξ. We classify all pairs $(\mathfrak{m},\xi)$ generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure $\mathfrak{m}$ is arbitrary, whereas the process ξ is stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of the Lebesgue measure, and ξ is essentially a Gaussian stationary increment process with linear drift. In the third, most interesting family, the measure $\mathfrak{m}$ has a density of the form αe^−λx, where α > 0, λ ∈ ℝ, whereas the process ξ is of the form ξ(t) = W(t) − λσ²(t) / 2 + c, where W is a zero-mean Gaussian process with stationary increments, σ²(t) = Var W(t), and c ∈ ℝ.

We develop necessary and sufficient conditions for uniqueness of the invariant measure of the filtering process associated to an ergodic hidden Markov model in a finite or countable state space. These results provide a complete solution to a problem posed by Blackwell (1957), and subsume earlier partial results due to Kaijser, Kochman and Reeds. The proofs of our main results are based on the stability theory of nonlinear filters.

By a theorem of Volkov [12] we know that on most graphs with positive probability the linearly vertex-reinforced random walk (VRRW) stays within a finite “trapping” subgraph at all large times. The question of whether this tail behavior occurs with probability one is open in general. In his thesis, Pemantle [5] proved, via a dynamical system approach, that for a VRRW on any complete graph the asymptotic frequency of visits is uniform over vertices. These techniques do not easily extend even to the setting of complete-like graphs, that is, complete graphs ornamented with finitely many leaves at each vertex. In this work we combine martingale and large deviation techniques to prove that almost surely the VRRW on any such graph spends positive (and equal) proportions of time on each of its nonleaf vertices. This behavior was previously shown to occur only up to event of positive probability (cf. Volkov [12]). We believe that our approach can be used as a building block in studying related questions on more general graphs. The same set of techniques is used to obtain explicit bounds on the speed of convergence of the empirical occupation measure.