Let $(B_n)$ be an increasing sequence of regions in $d$ -dimensional space with volume $n$ and with union $\mathbb{R}^d$. We prove a general central limit theorem for functionals of point sets, obtained either by restricting a homogeneous Poisson process to $(B_n)$, or by by taking $n$ uniformly distributed points in $(B_n)$. The sets $(B_n)$ could be all cubes but a more general class of regions$(B_n)$ is considered. Using this general result we obtain central limit theorems for specific functionals suchas total edge lengthand number of components, defined in terms of graphs such as the $k$-nearest neighbors graph, the sphere of influence graph and the Voronoi graph.

We characterize the limiting behavior of the number of nodes in level $k$ of binary search trees $T_n$ in the central region $1.2 \log n \leq 2.8 \log n$. Especially we show that the width $\bar{V}_n$ (the maximal number of internal nodes at the same level) satisfies $\bar{V}_n \sim (n/\sqrt{4\pi\log n})$ as $n \to \infty$ a.s.

We study the effects of boundary conditions in two-dimensional rigidity percolation. Specifically, we consider generic rigidity in the bond percolation model on the triangular lattice. We introduce a theory of boundary conditions and define two different notions of “rigid clusters,” called $\mathrm{r}^0$-clusters and $\mathrm{r}^1$-clusters, which correspond to free boundary conditions and wired boundary conditions respectively. The definition of an $\mathrm{r}^ 0$-cluster turns out to be equivalent to the definition of a rigid component used in earlier papers by Holroyd and Häggström. We define two critical probabilities, associated with the appearance of infinite $\mathrm{r}^0$-clusters and infinite $\mathrm{r}^1$-clusters respectively, and we prove that these two critical probabilities are in fact equal. Furthermore, we prove that for all parameter values $p$ except possibly this unique critical probability, the set of $\mathrm{r}^ 0$-clusters equals the set of $\mathrm{r}^ 1$-clusters almost surely. It is an open problem to determine what happens at the critical probability.

We consider a repulsion–attraction model for a random polymer of finite lengthin $\mathbb{Z}^d$. Its law is that of a finite simple random walk path in $\mathbb{Z}^d$ receiving a penalty $3^{-2\beta}$ for every self-intersection, and a reward $e^{\gamma/d}$ for every pair of neighboring monomers. The nonnegative parameters $\beta$ and $\gamma$ measure the strength of repellence and attraction, respectively.

We show that for $\gamma > \beta$ the attraction dominates the repulsion; that is, with high probability the polymer is contained in a finite box whose size is independent of the length of the polymer. For $\gamma<\beta$ the behavior is different. We give a lower bound for the rate at which the polymer extends in space. Indeed, we show that the probability for the polymer consisting of $n$ monomers to be contained in a cube of side length $\varepsilon n^{1/d}$ tends to zero as $n$ tends to infinity.

In dimension $d = 1$ we can carry out a finer analysis. Our main result is that for $0 < \gamma \leq \beta -1/2\log 2$ the end-to-end distance of the polymer grows linearly and a central limit theorem holds.

It remains open to determine the behavior for $\gamma \in (\beta - 1/2\log 2, \beta]$ .

We consider $\mathbb{Z}$ as an infinite lattice street where cars of integer length $m \geq 1$ can park. The parking process is described by a 0–1 interacting particle system such that a site $z \in \mathbb{Z}$ is in state 1 whenever a car has its rear end at z and 0 otherwise. Cars attempt to park after exponential times with parameter $\lambda$, leave after exponential times with parameter 1 and are not allowed to touch nor overlap. We define and study a jamming occupation density for this parking process, using the quasi-stationary distribution of a Markov chain related to the reversible measure of the particle system. An extension to a strip in $\mathbb{Z}^2$ is also investigated.

This paper studies stochastic particle approximations for Smoluchowski’s coagulation equation. A new stochastic algorithm with reduced variance is proposed. Its convergence behavior is investigated, when the number of simulation particles tends to infinity. Under appropriate assumptions on the coagulation kernel, the limit is the unique solution of the coagulation equation. Then detailed numerical experiments are performed, testing the applicability and efficiency of the algorithm. In particular, the gelation phenomenon (loss of mass in the coagulation equation) is studied numerically for several kernels. A striking feature of the new algorithm is a better convergence after the gelation point, providing a tool for detecting the mass of the gel.

A path-valued interacting particle systems model for the genealogical structure of genetic algorithms is presented. We connect the historical process and the distribution of the whole ancestral tree with a class of Feynman-Kac formulae on path space. We also prove increasing and uniform versions of propagation of chaos for appropriate particle block size and time horizon yielding what seems to be the first result of this type for this class of particle systems.

It has recently been observed that wide-sense self-similar processes have a rich linear structure analogous to that of wide-sense stationary processes. In this paper, a reproducing kernel Hilbert space (RKHS) approach is used to characterize this structure. The RKHS associated with a self-similar process on a variety of simple index sets has a straightforward description, provided that the scale-spectrum of the process can be factored. This RKHS description makes use of the Mellin transform and linear self-similar systems in much the same way that Laplace transforms and linear time-invariant systems are used to study stationary processes.

The RKHS results are applied to solve linear problems including projection, polynomial signal detection and polynomial amplitude estimation, for general wide-sense self-similar processes. These solutions are applied specifically to fractional Brownian motion (fBm). Minimum variance unbiased estimators are given for the amplitudes of polynomial trends in fBm, and two new innovations representations for fBm are presented.

W consider the class of autoregressive processes with ARCH(1)errors given by the stochastic difference equation

$$X_n = \alpha X_{n-1} + \sqrt{\beta + \lambda X_{n-1}^2}\varepsilon_n,\quad n \in \mathbb{N}$$

where $(\varepsilon_n)_{n \in \mathbb{N}$ are i.i.d random variables. Under general and tractable assumptions we show the existence and uniqueness of a stationary distribution. We prove that the stationary distribution has a Pareto-like tail with a well-specified tail index which depends on $\alpha, \lambda$ and the distribution of the innovations $(\varepsilon_n)_{n \in \mathbb{N}}$. This paper generalizes results for the ARCH(1) process (the case $\alpha = 0$). The generalization requires a new method of proof and we invoke a Tauberian theorem.

The exact convergence rates of the particle distributions in supercritical branching random walks and supercritical branching Wiener processes are obtained and a conjecture of Révész is confirmed.

A stochastic Lotka–Volterra model is formulated by using the semimartingale approach. The large deviation principle is established, and is used to obtain a bound for the asymptotics of the time to extinction of prey population. The bound is given in terms of past-dependent ODE’s closely related to the dynamics of the deterministic Lotka –Volterra model.

In this paper we prove the local sample path large deviation estimates for a general class of Markov chains with discontinuous statistics. The local rate function is represented in terms of the convergence parameter of associated local transform matrices. Our method is illustrated by the case of perturbated random walks in $\mathbb{Z}^d$.

We use McFadden’s integral equations for random RC filters to study the average distribution of Dubins–Freedman processes. These distributions are also stationary probability measures of Markov chains on [0,1], defined by the iteration of steps to the left $x \to ux$, and of steps to the right $x \to v + (1 - v)x$, where uand vare random from [0,1]. We establish new algorithms to compute the stationary measure of these chains.

Turning to specific examples, we show that, if the distributions of u and $1-v$ are Beta(a,1), or Beta (a, 2), or if u and $1 - v$ are the exponential of Gamma (a, 2) distributed random variables, then the stationary measure is a combination of various hypergeometric functions, which are often $_3 F_2$ functions. Our methods are based on a link that we establish between these Markov chains and some RC filters. We also determine the stationary distribution of RC filters in specific cases. These results generalize recent examples of Diaconis and Freedman.

In the context of a general multivariate financial market with transaction costs, we consider the problem of maximizing expected utility from terminal wealth. In contrast with the existing literature, where only the liquidation value of the terminal portfolio is relevant, we consider general utility functions which are only required to be consistent with the structure of the transaction costs. An important feature of our analysis is that the utility function is not required to be $C^1$. Such nonsmoothness is suggested by major natural examples. Our main result is an extension of the well-known dual formulation of the utility maximization problem to this context.

We study the distribution of steady-state queue lengths in multiclass queueing networks under a stable policy. We propose a general methodology based on Lyapunov functions for the performance analysis of infinite state Markov chains and apply it specifically to Markovian multiclass queueing networks. We establish a deeper connection between stability and performance of such networks by showing that if there exist linear or piecewise linear Lyapunov functions that show stability, then these Lyapunov functions can be used to establish geometric-type lower and upper bounds on the tail probabilities, and thus bounds on the expectation of the queue lengths. As an example of our results, for a reentrant line queueing network with two processing stations operating under a work-conserving policy, we show that $\mathrm{E}[L] = O(\frac{1}{1 - \rho^*)^2})$, where $L$ is the total number of customers in the system, and $\rho^*$ is the maximal actual or virtual traffic intensity in the network. In a Markovian setting, this extends a recent result by Dai and Vande Vate, which states that a reentrant line queueing network with two stations is globally stable if $\rho^*<1$. We also present several results on the performance of multiclass queueing networks operating under general Markovian and,in particular,priority policies. The results in this paper are the first that establish explicit geometric-type upper and lower bounds on tail probabilities of queue lengths for networks of such generality. Previous results provide numerical bounds and only on the expectation, not the distribution, of queue lengths.