We consider a two-dimensional catalytic surface reaction between X and $Y_n$ with $Y_n + nX \to nXY$, where $Y_n$ is a polymer consisting of n identical atoms, each denoted by Y, and X is a monomer. The reactants X and $Y_n$ are present above the surface in a gaseous phase, and bond to the surface at certain rates. The resulting atoms X and Y on the surface react if they are sufficiently close to each other; the product XY then leaves the surface. A classical example is the oxidation of carbon monoxide on a platinum surface. In this case, $n = 2, X = CO$ and $Y_2 = O_2$. We consider the case in which the polymer consists of $n = N^2$ atoms, arranged in a square of length N, with N large. We show that when the range of interaction is large compared to the polymer size, X and Y will typically coexist on the catalytic surface for appropriate bonding rates. If, however, the range of interaction is small compared to the polymer size, then, irrespective of the bonding rates, the surface will eventually be fully occupied by the monomer X.

A Poisson-Voronoi tessellation (PVT) is a tiling of the Euclidean plane in which centers of individual tiles constitute a Poisson field and each tile comprises the locations that are closest to a given center with respect to a prescribed norm. Many spatial systems in which rare, randomly distributed centers compete for space should be well approximated by a PVT. Examples that we can handle rigorously include multitype threshold vote automata, in which $\kappa$ different camps compete for voters stationed on the two-dimensional lattice. According to the deterministic, discrete-time update rule, a voter changes affiliation only to that of a unique opposing camp having more than $\theta$ representatives in the voter's neighborhood. We establish a PVT limit for such dynamics started from completely random configurations, as the number of camps becomes large, so that the density of initial "pockets of consensus" tends to 0. Our methods combine nucleation analysis, Poisson approximation, and shape theory.

Throw a straight line at random into a plane, within which is inscribed a square grid. Color black each grid vertex that lies above the line, and white each vertex below it. Now remove the line, and attempt to reconstruct it from the pattern of vertex colors in an $m \times m$ section of the grid. We show that for all $\varepsilon > 0$, the line can be approximated to within order $m^{-1}(\log m)(\log \log m)^{1+\varepsilon}$, with probability 1, and that there is no deterministic subsequence along which the best achievable rate is better than $(m \log m)^{-1}(\log \log m)^{-1-\varepsilon}$ with positive probability. Both these results fail if $\varepsilon = 0$. More generally, we provide a complete characterization of almost sure rates of approximation, in terms of the convergence or divergence of an infinite series. Applying these results, we develop near-optimal local linear approximations to general smooth boundaries. We address the case where vertex color is a shade of gray, varying in the continuum and subject to stochastic error.

A new and unified approach is presented for constructing joint generating functions for quantities of interest associated with pattern formation in binary sequences. The method is based on results on exponential families of Markov chains. The tools we use are not only new for this area; they seem to be the right approach for deriving general explicit distributional results.

The competitive learning vector quantization (CLVQ) algorithm with constant step $\varepsilon > 0$--also known as the Kohonen algorithm with 0 neighbors--is studied when the stimuli are i.i.d. vectors. Its first noticeable feature is that, unlike the one-dimensional case which has $n!$ absorbing subsets, the CLVQ algorithm is "irreducible on open sets" whenever the stimuli distribution has a path-connected support with a nonempty interior. Then the Doeblin recurrence (or uniform ergodicity) of the algorithm is established under some convexity assumption on the support. Several properties of the invariant probability measure $\nu^{\varepsilon}$ are studied, including support location and absolute continuity with respect to the Lebesgue measure. Finally, the weak limit set of $\nu^{\varepsilon}$ as $\varepsilon \to 0$ is investigated.

We describe a class of adaptive algorithms for approximating the global minimum of a continuous function on the unit interval. The limiting distribution of the error is derived under the assumption of Wiener measure on the objective functions. For any $\delta > 0$, we construct an algorithm which has error converging to zero at rate $n^{(-1-\delta)}$. in the number of function evaluations n. This convergence rate contrasts with the $n^{-1/2}$ rate of previously studied nonadaptive methods.

A guiding principle in the efficient estimation of rare-event probabilities by Monte Carlo is that importance sampling based on the change of measure suggested by a large deviations analysis can reduce variance by many orders of magnitude. In a variety of settings, this approach has led to estimators that are optimal in an asymptotic sense. We give examples, however, in which importance sampling estimators based on a large deviations change of measure have provably poor performance. The estimators can have variance that decreases at a slower rate than a naive estimator, variance that increases with the rarity of the event, and even infinite variance. For each example, we provide an alternative estimator with provably efficient performance. A common feature of our examples is that they allow more than one way for a rare event to occur; our alternative estimators give explicit weight to lower probability paths neglected by leading-term asymptotics.

Brownian networks are a class of linear stochastic control systems that arise as heavy traffic approximations in queueing theory. Such Brownian system models have been used to approximate problems of dynamic routing, dynamic sequencing and dynamic input control for queueing networks. A number of specific examples have been analyzed in recent years, and in each case the Brownian network has been successfully reduced to an "equivalent workload formulation" of lower dimension. In this article we explain that reduction in general terms, using an orthogonal decomposition that distinguishes between reversible and irreversible controls.

Using the ergodic theory of nonnegative matrices, conditions are obtained for the $L^2$ and almost sure convergence of a supercritical multitype branching process with varying environment, normed by its mean. We also give conditions for the extinction probability of the limit to equal that of the process.

The theory developed allows for different types to grow at different rates, and an example of this is given, taken from the construction of a spatially inhomogeneous diffusion on the Sierpinski gasket.

A Poisson point process $\Psi$ in d-dimensional Euclidean space and in time is used to generate a birth-growth model: seeds are born randomly at locations $x_i$ in $\mathbb{R}^d$ at times $t_i \epsilon [0, \infty)$. Once a seed is born, it begins to create a cell by growing radially in all directions with speed $v > 0$. Points of $\Psi$ contained in such cells are discarded, that is, thinned.We study the asymptotic distribution of the number of seeds in a region, as the volume of the region tends to infinity. When $d = 1$, we establish conditions under which the evolution over time of the number of seeds in a region is approximated by a Wiener process. When $d \geq 1$, we give conditions for asymptotic normality. Rates of convergence are given in all cases.

For an infinite sequence of independent and identically distributed (i.i.d.) random variables, the k-record process consists of those terms that are the kth largest at their appearance. Ignatov's theorem states that the k-record processes, $k = 1, 2, \dots ,$ are i.i.d. A new proof is given which is based on a "continualization" argument. An advantage of this fairly simple approach is that Ignatov's theorem can be stated in a more general form by allowing for different tiebreaking rules. In particular, three tiebreakers are considered and shown to be related to Bernoulli, geometric and Poisson distributions.

A continuous time Markov chain is observed with Gaussian white noise added to it. To the well-known problem of continuously estimating the current state of the chain, we introduce the additional option of continuously varying the sampling rates, as long as some restriction (or cost) on the average sampling rate is satisfied. The optimal solution to this "dynamic sampling" problem is presented and analyzed in closed form for the two-state symmetric case. It is shown that the resulting dynamic sampling procedure has a much lower asymptotic average error rate compared to the one obtained when sampling at a constant rate. Alternatively, the dynamic sampling procedure can provide the same error rate using a much lower average sampling rate. The relative efficiency of the dynamic sampling procedure may in fact tend to infinity in some extreme cases.