On asymptotic constants in the theory of extremes for Gaussian processes

This paper gives a new representation of Pickands' constants, which arise in the study of extremes for a variety of Gaussian processes. Using this representation, we resolve the long-standing problem of devising a reliable algorithm for estimating these constants. A detailed error analysis illustrates the strength of our approach.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ534 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Stochastic billiards for sampling from the boundary of a convex set

Stochastic billiards can be used for approximate sampling from the boundary of a bounded convex set through the Markov Chain Monte Carlo (MCMC) paradigm. This paper studies how many steps of the underlying Markov chain are required to get samples (approximately) from the uniform distribution on the boundary of the set, for sets with an upper bound on the curvature of the boundary. Our main theorem implies a polynomial-time algorithm for sampling from the boundary of such sets

Randomized longest-queue-first scheduling for large-scale buffered systems

We develop diffusion approximations for parallel-queueing systems with the randomized longest-queue-first scheduling algorithm by establishing new mean-field limit theorems as the number of buffers $n\to\infty$. We achieve this by allowing the number of sampled buffers $d=d(n)$ to depend on the number of buffers $n$, which yields an asymptotic `decoupling' of the queue length processes. We show through simulation experiments that the resulting approximation is accurate even for moderate values of $n$ and $d(n)$. To our knowledge, we are the first to derive diffusion approximations for a queueing system in the large-buffer mean-field regime. Another noteworthy feature of our scaling idea is that the randomized longest-queue-first algorithm emulates the longest-queue-first algorithm, yet is computationally more attractive. The analysis of the system performance as a function of $d(n)$ is facilitated by the multi-scale nature in our limit theorems: the various processes we study have different space scalings. This allows us to show the trade-off between performance and complexity of the randomized longest-queue-first scheduling algorithm

Large deviations for random walks under subexponentiality: the big-jump domain

For a given one-dimensional random walk $\{S_n\}$ with a subexponential step-size distribution, we present a unifying theory to study the sequences $\{x_n\}$ for which $\mathsf{P}\{S_n>x\}\sim n\mathsf{P}\{S_1>x\}$ as $n\to\infty$ uniformly for $x\ge x_n$. We also investigate the stronger "local" analogue, $\mathsf{P}\{S_n\in(x,x+T]\}\sim n\mathsf{P}\{S_1\in(x,x+T]\}$. Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory. When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.Comment: Published in at http://dx.doi.org/10.1214/07-AOP382 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Reflected Brownian motion in a wedge: sum-of-exponential stationary densities

We give necessary and sufficient conditions for the stationary density of semimartingale reflected Brownian motion in a wedge to be written as a finite sum of terms of exponential product form. Relying on geometric ideas reminiscent of the reflection principle, we give an explicit formula for the density in such cases

Sensitivity analysis for diffusion processes constrained to an orthant

This paper studies diffusion processes constrained to the positive orthant under infinitesimal changes in the drift. Our first main result states that any constrained function and its (left) drift-derivative is the unique solution to an augmented Skorohod problem. Our second main result uses this characterization to establish a basic adjoint relationship for the stationary distribution of the constrained diffusion process jointly with its left-derivative process.Comment: Published in at http://dx.doi.org/10.1214/13-AAP967 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org