Under very mild conditions, we prove that the number of components in a decomposable logarithmic combinatorial structure has a distribution which is close to Poisson in total variation. The conditions are satisfied for all assemblies, multisets and selections in the logarithmic class.The error in the Poisson approximation is shown under marginally more restrictive conditions to be of exact order $O(1/\log n)$, by exhibiting the penultimate asymptotic approximation; similar results have previously been obtained by Hwang [20], under stronger assumptions.Our method is entirely probabilistic, and the conditions can readily be verified in practice.

In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising from the recent advent of methods of “perfect simulation”: it helps to establish the “price of perfection” for such methods. In general, one can always achieve efficient coupling if the coupling is allowed to “cheat”(if each component’s behavior is affected by the future behavior of the other component), but the situation is more interesting if the coupling is required to be co-adapted. We present an informal heuristic for the existence of an efficient coupling, and justify the heuristic by proving rigorous results and examples in the contexts of finite reversible Markov chains and of reflecting Brownian motion in planar domains.

Consider a process in which information is transmitted from a given root node on a noisy tree network $T$.We start with an unbiased random bit $R$ at the root of the tree and send it down the edges of $T$.On every edge the bit can be reversed with probability $\varepsilon$, and these errors occur independently. The goal is to reconstruct $R$ from the values which arrive at the $n$th level of the tree. This model has been studied in information theory,genetics and statistical mechanics.We bound the reconstruction probability from above, using the maximum flow on $T$ viewed as a capacitated network, and from below using the electrical conductance of $T$. For general infinite trees, we establish a sharp threshold: the probability of correct reconstruction tends to 1/2 as $n \to \infty$ if $(1 - 2\varepsilon)^2 < p_c(T)$, but the reconstruction probability stays bounded away from ½ if the opposite inequality holds. Here $p_c(T)$ is the critical probability for percolation on $T$; in particular $p_c(T) = 1/b$ for the $b + 1$-regular tree. The asymptotic reconstruction problem is equivalent to purity of the “free boundary” Gibbs state for the Ising model on a tree. The special case of regular trees was solved in 1995 by Bleher, Ruiz and Zagrebnov; our extension to general trees depends on a coupling argument and on a reconstruction algorighm that weights the input bits by the electrical current flow from the root to the leaves.

We consider a two-dimensional Kac equation without cutoff,which we relate to a stochastic differential equation.We prove the existence of a solution for this SDE, and we use the Malliavin calculus (or stochastic calculus of variations) to prove that the law of this solution admits a smooth density with respect to the Lebesgue measure on $\mathbf{R}^2$.This density satisfies the Kac equation.

We consider a random surface $\Phi$ in $\mathbb{R}^d$ tessellating the space into cells and a random vector field $u$ which is smooth on each cell but may jump on $\Phi$. Assuming the pair $(\Phi, u)$ stationary we prove a relationship between the stationary probability measure $P$ and the Palm probability measure $P_{\Phi}$ of $P$ with respect to the random surface measure associated with $\Phi$. This result involves the flow of $u$ induced on the individual cells and generalizes a well-known inversion formula for stationary point processes on the line. An immediate consequence of this result is a formula for certain generalized contact distribution functions of $\Phi$, and as first application we prove a result on the spherical contact distribution in stochastic geometry. As another application we prove an invariance property for $P_{\Phi}$ which again generalizes a corresponding property in dimension $d = 1$. Under the assumption that the flow can be defined for all time points, we consider the point process $N$ of sucessive crossing times starting in the origin 0. If the flow is volume preserving, then $N$ is stationary and we express its Palm probability in terms of $P_{\Phi}$.

This paper is concerned with the simulation of a 2-dimensional stochastic differential equation motivated by some physical phenomena of fluid mechanics. The drift and diffusion coefficients of the equation admitting local singularities, we are led to study a particular term of strong perturbation denoted by “Brownian impulse.” Our suggestion for the simulation is to replace the singularity by a jump on which our study therefore focuses.

Given an ordering of the vertices of a finite graph, let the induced weight for an edge be the separation of its endpoints in the ordering. Layout problems involve choosing the ordering to minimize a cost functional such as the sum or maximum of the edge weights. We give growth rates for the costs of some of these problems on supercritical percolation processes and supercritical random geometric graphs, obtained by placing vertices randomly in the unit cube and joining them whenever at most some threshold distance apart.

A packing of a collection of rectangles contained in $[0, 1]^2$ is a disjoint subcollection; the wasted space is the measure of the area of the part of $[0, 1]^2$ not covered by the subcollection.A simple packing has the further restriction that each vertical line meets at most one rectangle of the packing. Given a collection of $N$ independent uniformly distributed subrectangles of $[0, 1]$, we proved in a previous work that there exists a number $K$ such that the wasted space $W_N$ in an optimal simple packing of these rectangles satisfies for all $N$

EW_N \leq \frac{K}{\sqrt{N}} \exp K\sqrt{\log N}.

We prove here that

\frac{1}{K\sqrt{N}} \exp \frac{1}{K} \sqrt{N} \leq EW_N.

This work is concerned with aggregations in a singularly perturbed Markov chain having a finite state space and fast and slow motions.The state space of the underlying Markov chain can be decomposed into several groups of recurrent states and a group of transient states.The asymptotic properties are studied through sequences of unscaled and scaled occupation measures.By treating the states within each recurrent class as a single state, an aggregated process is defined and shown to be convergent to a limit Markov chain.In addition, it is shown that a sequence of suitably rescaled occupation measures converges to a switching diffusion process weakly.

We consider the number of overlappingoccurrences up to a fixed time of one or several “rare ”patterns in a stationary finite-state Markov chain. We derive a bound for the total variation distance between the distribution of this quantity and a compound Poisson distribution, using general results on compound Poisson approximation for Markov chains by Erhardsson. If the state space is $\{0, 1\}$ and the pattern is a head run ($111 \dots 111$), the bound is completely explicit and improves on an earlier bound given by Geske, Godbole, Schaffner, Skolnick and Wallstrom. In general, the bound can be computed by solving five linear equation systems of dimension at most the number of states plus the sum of the lengths of the patterns. We also give approximations with error bounds for the distributions of the first occurrence time of a head run of fixed length and the longest head run occurringup to a fixed time. Finally, we consider the sojourn time in an “extreme” subset of the state space by a stationary birth–death chain and derive a bound for the total variation distance between the distribution of this quantity and a compound Poisson distribution.

We consider an $M/M/infty$ queue with servers ranked as ${1, 2, 3,\dots}$. The Poisson arrival stream has rate $\lambda$ and each server works at rate $\mu$. A new arrival takes the lowest ranked available server.We let $S$ be the set of occupied servers and $|S|$ is the number of elements of $S$.We study the distribution of max $(S)$ in the asymptotic limit of $\rho = \lambda/\mu \to \infty$.Setting $P(m) = \Pr[max(S) > m]$ we find that the asymptotic structure of the problem is different according as $m = O(1)$ or $m \to \infty$, at the same rate as $\rho$. For the latter it is furthermore necessary to distinguish the cases $m/\rho < 1, m/\rho \approx 1$ and $m/\rho > 1$.We also estimate the average amount of wasted stor- age space, which is defined by $E(max(S)) - \rho$. This is the average number of idle servers that are ranked below the maximum occupied one.We also relate our results to those obtained by probabilistic approaches. Numerical studies demonstrate the accuracy of the asymptotic results.

We analyze the pathwise approximation for systems of stochastic differential equations.The pathwise distance between the solution and its approximation is measured globally on the unit interval in the $L_{\infty}$-norm, and we study the expectation of this distance. For systems with additive noise we obtain sharp lower and upper bounds for the minimal error in the class of arbitrary methods which use discrete observations of a Brownian path. The optimal order is achieved by an Euler scheme with adaptive step-size control. We illustrate the superiority of the adaptive method compared to equidistant discretization by a simulation experiment.

We study a generalization of the arc-sine law. In particular we provide new results about the distribution of the time spent by a BM with drift inside a band, giving the Laplace transform of the characteristic function. If one of the extremes of the band goes to infinity, our formula agrees with the results given in Akahori and Takàcs.We apply these results to the pricing of exotic option contracts known as corridor derivatives.We then discuss the inversion problem comparing different numerical methods.

It is common to have interval predictions for volatilities and other quantities governing securities prices. The purpose of this paper is to provide an exact method for converting such intervals into arbitrage based prices of financial derivatives or industrial or contractual options.We call this procedure conservative delta hedging. The proposed approach will permit an institution’s management a greater oversight of its exposure to risk.