Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset\mathbb{R}^{d}$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,\ldots,d\}$, customarily denoted by $\mathcal{L}(V_{C})$. The aim of the present paper is to provide a Berry–Esseen bound for the normal approximation of $\mathcal{L}(V_{C})$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_{n}})$, $n\geq1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein’s method and second-order Poincaré inequality, (3) concentration estimates and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp [Inf. Inference 3 (2014) 224–294] and McCoy and Tropp [Discrete Comput. Geom. 51 (2014) 926–963] about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.

In wireless networks, where each node transmits independently of other nodes in the network (the ALOHA protocol), the expected delay experienced by a packet until it is successfully received at any other node is known to be infinite for the signal-to-interference-plus-noise-ratio (SINR) model with node locations distributed according to a Poisson point process. Consequently, the information velocity, defined as the limit of the ratio of the distance to the destination and the time taken for a packet to successfully reach the destination over multiple hops, is zero, as the distance tends to infinity. A nearest neighbor distance based power control policy is proposed to show that the expected delay required for a packet to be successfully received at the nearest neighbor can be made finite. Moreover, the information velocity is also shown to be nonzero with the proposed power control policy. The condition under which these results hold does not depend on the intensity of the underlying Poisson point process.

We consider a stochastic particle system in which a finite number of particles interact with one another via a common energy tank. Interaction rate for each particle is proportional to the square root of its kinetic energy, as is consistent with analogous mechanical models. Our main result is that the rate of convergence to equilibrium for such a system is ${\sim}t^{-2}$, more precisely it is faster than a constant times $t^{-2+\varepsilon}$ for any $\varepsilon>0$. A discussion of exponential vs. polynomial convergence for similar particle systems is included.

We study a percolation model on the square lattice, where clusters “freeze” (stop growing) as soon as their volume (i.e., the number of sites they contain) gets larger than $N$, the parameter of the model. A model where clusters freeze when they reach diameter at least $N$ was studied in van den Berg, de Lima and Nolin [Random Structures Algorithms 40 (2012) 220–226] and Kiss [Probab. Theory Related Fields 163 (2015) 713–768]. Using volume as a way to measure the size of a cluster—instead of diameter—leads, for large $N$, to a quite different behavior (contrary to what happens on the binary tree van den Berg, Kiss and Nolin [Electron. Commun. Probab. 17 (2012) 1–11], where the volume model and the diameter model are “asymptotically the same”). In particular, we show the existence of a sequence of “exceptional” length scales.

We consider a system of independent one-dimensional random walkers where new particles are added at the origin at fixed rate whenever there is no older particle present at the origin. A Poisson ansatz leads to a semi-linear lattice heat equation and predicts that starting from the empty configuration the total number of particles grows as $c\sqrt{t}\log t$. We confirm this prediction and also describe the asymptotic macroscopic profile of the particle configuration.

We consider one-dimensional, boundary driven, weakly asymmetric exclusion processes in contact with reservoirs at fixed density. For a general set of initial measures and by using a microscopic Cole–Hopf transformation, we derive the nonequilibrium fluctuations which are given by a generalized Ornstein–Uhlenbeck process.

A recent conjecture of Fyodorov–Hiary–Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is $\exp \{\log \log T-\frac{3}{4}\log \log \log T+O(1)\}$, for an interval at (large) height $T$. In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.

In this paper, we propose tight upper and lower bounds for the Wasserstein distance between any two univariate continuous distributions with probability densities $p_{1}$ and $p_{2}$ having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio $p_{1}/p_{2}$ as well as the Stein kernel $\tau_{1}$ of $p_{1}$. The method of proof relies on a new variant of Stein’s method which manipulates Stein operators.

We give several applications of these bounds. Our main application is in Bayesian statistics: we derive explicit data-driven bounds on the Wasserstein distance between the posterior distribution based on a given prior and the no-prior posterior based uniquely on the sampling distribution. This is the first finite sample result confirming the well-known fact that with well-identified parameters and large sample sizes, reasonable choices of prior distributions will have only minor effects on posterior inferences if the data are benign.

Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_{0}$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $n\to\infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{S(t);t\ge0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the start of the regenerative cycle. Properties of the process $S$, including the jump size and stationary distributions, are determined.

Consider a multidimensional diffusion process $X=\{X(t):t\in [0,1]\}$. Let $\varepsilon>0$ be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of $X$, we construct a probability space, supporting both $X$ and an explicit, piecewise constant, fully simulatable process $X_{\varepsilon}$ such that

\[\sup_{0\leq t\leq1}\Vert X_{\varepsilon}(t)-X(t)\Vert_{\infty}<\varepsilon\] with probability one. Moreover, the user can adaptively choose $\varepsilon^{\prime}\in (0,\varepsilon )$ so that $X_{\varepsilon^{\prime}}$ (also piecewise constant and fully simulatable) can be constructed conditional on $X_{\varepsilon}$ to ensure an error smaller than $\varepsilon^{\prime}$ with probability one. Our construction requires a detailed study of continuity estimates of the Itô map using Lyons’ theory of rough paths. We approximate the underlying Brownian motion, jointly with the Lévy areas with a deterministic $\varepsilon$ error in the underlying rough path metric.

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (nonnormal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erdős and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic “puzzle graph” by using connectivity properties of a random “people graph” on the same set of vertices. We presume the Erdős–Rényi people graph with edge probability $p$ and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with $N$ vertices of degrees about $D$ (in the appropriate sense), this probability is close to 1 or small depending on whether $pD\log N$ is large or small. The one dimensional ring and two dimensional torus puzzles are studied in more detail and in many cases the exact scaling of the critical probability is obtained. The paper strengthens several results of Brummitt, Chatterjee, Dey, and Sivakoff who introduced this model.

In this paper, we study the asymptotics of the degree sequence of permutation graphs associated with a sequence of random permutations. The limiting finite-dimensional distributions of the degree proportions are established using results from graph and permutation limit theories. In particular, we show that for a uniform random permutation, the joint distribution of the degree proportions of the vertices labeled $\lceil nr_{1}\rceil,\lceil nr_{2}\rceil,\ldots,\lceil nr_{s}\rceil$ in the associated permutation graph converges to independent random variables $D(r_{1}),D(r_{2}),\ldots,D(r_{s})$, where $D(r_{i})\sim\operatorname{Unif}(r_{i},1-r_{i})$, for $r_{i}\in[0,1]$ and $i\in\{1,2,\ldots,s\}$. Moreover, the degree proportion of the mid-vertex (the vertex labeled $n/2$) has a central limit theorem, and the minimum degree converges to a Rayleigh distribution after an appropriate scaling. Finally, the asymptotic finite-dimensional distributions of the permutation graph associated with a Mallows random permutation is determined, and interesting phase transitions are observed. Our results extend to other nonuniform measures on permutations as well.

We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer [Ann. Appl. Probab. 9 (1999) 904–950] for optimal expected utility from nonnegative random variables to the level of optimal expected utility from increasing, adapted controls. The main contributions are the formulation of a suitable duality framework, the identification of the problem’s dual functional as well as the full duality for the primal and dual value functions and their optimizers. The scope of our results is illustrated by an irreversible investment problem and the Hindy–Huang–Kreps utility maximization problem for incomplete financial markets.

We consider two stationary versions of the Eden model, on the upper half planar lattice, resulting in an infinite forest covering the half plane. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. Using the mass transport principle, we generalize the result to Eden model in graphs of the form $G\times\mathbb{Z}_{+}$, where $G$ is a Cayley graph. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Häggström in 2007 [Ann. Appl. Probab. 17 (2007) 1639–1656].

We consider $M/\mathit{Ph}/n+M$ queueing systems in steady state. We prove that the Wasserstein distance between the stationary distribution of the normalized system size process and that of a piecewise Ornstein–Uhlenbeck (OU) process is bounded by $C/\sqrt{\lambda}$, where the constant $C$ is independent of the arrival rate $\lambda$ and the number of servers $n$ as long as they are in the Halfin-Whitt parameter regime. For each integer $m>0$, we also establish a similar bound for the difference of the $m$th steady-state moments. For the proofs, we develop a modular framework that is based on Stein’s method. The framework has three components: Poisson equation, generator coupling, and state space collapse. The framework, with further refinement, is likely applicable to steady-state diffusion approximations for other stochastic systems.

Given an instance of the preferential attachment graph $G_{n}=([n],E_{n})$, we would like to find vertex 1, using only “local” information about the graph; that is, by exploring the neighborhoods of small sets of vertices. Borgs et al. gave an algorithm which runs in time $O(\log^{4}n)$, which is local in the sense that at each step, it needs only to search the neighborhood of a set of vertices of size $O(\log^{4}n)$. We give an algorithm to find vertex 1, which w.h.p. runs in time $O(\omega\log n)$ and which is local in the strongest sense of operating only on neighborhoods of single vertices. Here $\omega=\omega(n)$ is any function that goes to infinity with $n$.

Determining the mixing time of Kac’s random walk on the sphere $\mathrm{S}^{n-1}$ is a long-standing open problem. We show that the total variation mixing time of Kac’s walk on $\mathrm{S}^{n-1}$ is between $\frac{1}{2}n\log(n)$ and $200n\log(n)$ for all $n$ sufficiently large. Our bound is thus optimal up to a constant factor, improving on the best-known upper bound of $O(n^{5}\log(n)^{2})$ due to Jiang [Ann. Appl. Probab. 22 (2012) 1712–1727]. Our main tool is a “non-Markovian” coupling recently introduced by the second author in [Ann. Appl. Probab. 24 (2014) 114–130] for obtaining the convergence rates of certain high dimensional Gibbs samplers in continuous state spaces.