This paper studies rescaled images, under $\exp^{-1}_{\mu}$, of the sample Fréchet means of i.i.d. random variables $\{X_{k}\vert k\geq 1\}$ with Fréchet mean $\mu$ on a Riemannian manifold. We show that, with appropriate scaling, these images converge weakly to a diffusion process. Similar to the Euclidean case, this limiting diffusion is a Brownian motion up to a linear transformation. However, in addition to the covariance structure of $\exp^{-1}_{\mu}(X_{1})$, this linear transformation also depends on the global Riemannian structure of the manifold.

Isotropic Gaussian random fields on the sphere are characterized by Karhunen–Loève expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample Hölder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen–Loève expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated. The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated. Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven.

In this article, we introduce the parametrix technique in order to construct fundamental solutions as a general method based on semigroups and their generators. This leads to a probabilistic interpretation of the parametrix method that is amenable to Monte Carlo simulation. We consider the explicit examples of continuous diffusions and jump driven stochastic differential equations with Hölder continuous coefficients.

We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring, the particles are annihilated. This is a nonmonotone model, which makes the analysis more difficult.

We consider the extinction window of this model in the finite mean-field case, where there are $n$ sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic.

We study the effects of fast spatial movement of molecules on the dynamics of chemical species in a spatially heterogeneous chemical reaction network using a compartment model. The reaction networks we consider are either single- or multi-scale. When reaction dynamics is on a single-scale, fast spatial movement has a simple effect of averaging reactions over the distribution of all the species. When reaction dynamics is on multiple scales, we show that spatial movement of molecules has different effects depending on whether the movement of each type of species is faster or slower than the effective reaction dynamics on this molecular type. We obtain results for both when the system is without and with conserved quantities, which are linear combinations of species evolving only on the slower time scale.

This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first consider the case of compound Poisson (possibly Markov modulated) input. In this case, we analyze the complexity of our procedure as the dimension of the network increases and show that, under certain assumptions, the algorithm has polynomial-expected termination time. Our methodology includes procedures that are of interest beyond steady-state simulation and reflected processes. For instance, we use wavelets to construct a piecewise linear function that can be guaranteed to be within $\varepsilon$ distance (deterministic) in the uniform norm to Brownian motion in any compact time interval.

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical reflection principle to work. We call this method a weak reflection principle and show that it provides solutions to many problems for which the classical reflection principle is typically used. In addition, unlike the classical reflection principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak reflection principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the Lévy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak reflection principle in financial mathematics, computational methods and inverse problems.

The random geometric graph is obtained by sampling $n$ points from the unit square (uniformly at random and independently), and connecting two points whenever their distance is at most $r$, for some given $r=r(n)$. We consider the following variation on the random geometric graph: in each of $n$ rounds in total, a player is offered two random points from the unit square, and has to select exactly one of these two points for inclusion in the evolving geometric graph.

We study the problem of avoiding a linear-sized (or “giant”) component in this setting. Specifically, we show that for any $r\ll(n\log\log n)^{-1/3}$ there is a strategy that succeeds in keeping all component sizes sublinear, with probability tending to one as $n\to\infty$. We also show that this is tight in the following sense: for any $r\gg(n\log\log n)^{-1/3}$, the player will be forced to create a component of size $(1-o(1))n$, no matter how he plays, again with probability tending to one as $n\to\infty$. We also prove that the corresponding offline problem exhibits a similar threshold behaviour at $r(n)=\Theta(n^{-1/3})$.

These findings should be compared to the existing results for the (ordinary) random geometric graph: there a giant component arises with high probability once $r$ is of order $n^{-1/2}$. Thus, our results show, in particular, that in the geometric setting the power of choices can be exploited to a much larger extent than in the classical Erdős–Rényi random graph, where the appearance of a giant component can only be delayed by a constant factor.

There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a random point process in $\mathbb{R}^{d}$, and the edges and faces are determined according to some deterministic rule, typically leading to Čech and Vietoris–Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either i.i.d. observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction or repulsion, we find phenomena quantitatively different from those observed in the i.i.d. and Poisson cases. From the point of view of topological data analysis, our results seriously impact considerations of model (non)robustness for statistical inference. Our proofs rely on analysis of subgraph and component counts of stationary point processes, which are of independent interest in stochastic geometry.

We develop a heavy traffic diffusion limit theorem under nonstandard spatial scaling for the queue length process in a single server queue employing shortest remaining processing time (SRPT). For processing time distributions with unbounded support, it has been shown that standard diffusion scaling yields an identically zero limit. We specify an alternative spatial scaling that produces a nonzero limit. Our model allows for renewal arrivals and i.i.d. processing times satisfying a rapid variation condition. We add a corrective spatial scale factor to standard diffusion scaling, and specify conditions under which the sequence of unconventionally scaled queue length processes converges in distribution to the same nonzero reflected Brownian motion to which the sequence of conventionally scaled workload processes converges. Consequently, this corrective spatial scale factor characterizes the order of magnitude difference between the queue length and workload processes of SRPT queues in heavy traffic. It is determined by the processing time distribution such that the rate at which it tends to infinity depends on the rate at which the tail of the processing time distribution tends to zero. For Weibull processing time distributions, we restate this result in a manner that makes the resulting state space collapse more apparent.

We study a Wiener disorder problem of detecting the minimum of $N$ change-points in $N$ observation channels coupled by correlated noises. It is assumed that the observations in each dimension can have different strengths and that the change-points may differ from channel to channel. The objective is the quickest detection of the minimum of the $N$ change-points. We adopt a min–max approach and consider an extended Lorden’s criterion, which is minimized subject to a constraint on the mean time to the first false alarm. It is seen that, under partial information of the post-change drifts and a general nonsingular stochastic correlation structure in the noises, the minimum of $N$ cumulative sums (CUSUM) stopping rules is asymptotically optimal as the mean time to the first false alarm increases without bound. We further discuss applications of this result with emphasis on its implications to the efficiency of the decentralized versus the centralized systems of observations which arise in engineering.

In this paper, we consider two attractive stochastic spatial models in which each site can be in state 0, 1 or 2: Krone’s model in which 0${}={}$vacant, 1${}={}$juvenile and 2${}={}$a mature individual capable of giving birth, and the Staver–Levin forest model in which 0${}={}$grass, 1${}={}$sapling and 2${}={}$tree. Our first result shows that if $(0,0)$ is an unstable fixed point of the mean-field ODE for densities of 1’s and 2’s then when the range of interaction is large, there is positive probability of survival starting from a finite set and a stationary distribution in which all three types are present. The result we obtain in this way is asymptotically sharp for Krone’s model. However, in the Staver–Levin forest model, if $(0,0)$ is attracting then there may also be another stable fixed point for the ODE, and in some of these cases there is a nontrivial stationary distribution.

We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdős–Rényi graph on $N$ vertices and with connection probability $p_{0}$; under the alternative, there is an unknown subgraph on $n$ vertices where the connection probability is $p_{1}>p_{0}$. In Arias-Castro and Verzelen [Ann. Statist. 42 (2014) 940–969], we focused on the asymptotically dense regime where $p_{0}$ is large enough that $np_{0}>(n/N)^{o(1)}$. We consider here the asymptotically sparse regime where $p_{0}$ is small enough that $np_{0}<(n/N)^{c_{0}}$ for some $c_{0}>0$. As before, we derive information theoretic lower bounds, and also establish the performance of various tests. Compared to our previous work [Ann. Statist. 42 (2014) 940–969], the arguments for the lower bounds are based on the same technology, but are substantially more technical in the details; also, the methods we study are different: besides a variant of the scan statistic, we study other tests statistics such as the size of the largest connected component, the number of triangles, and the number of subtrees of a given size. Our detection bounds are sharp, except in the Poisson regime where we were not able to fully characterize the constant arising in the bound.

We study a dynamic scheduling problem for a multi-class queueing network with a large pool of statistically identical servers. The arrival processes are Poisson, and service times and patience times are assumed to be exponentially distributed and class dependent. The optimization criterion is the expected long time average (ergodic) of a general (nonlinear) running cost function of the queue lengths. We consider this control problem in the Halfin–Whitt (QED) regime, that is, the number of servers $n$ and the total offered load $\mathbf{r}$ scale like $n\approx\mathbf{r}+\hat{\rho}\sqrt{\mathbf{r}}$ for some constant $\hat{\rho}$. This problem was proposed in [Ann. Appl. Probab. 14 (2004) 1084–1134, Section 5.2].

The optimal solution of this control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. We introduce a broad class of ergodic control problems for controlled diffusions, which includes a large class of queueing models in the diffusion approximation, and establish a complete characterization of optimality via the study of the associated HJB equation. We also prove the asymptotic convergence of the values for the multi-class queueing control problem to the value of the associated ergodic diffusion control problem. The proof relies on an approximation method by spatial truncation for the ergodic control of diffusion processes, where the Markov policies follow a fixed priority policy outside a fixed compact set.

Extreme value theory is part and parcel of any study of order statistics in one dimension. Our aim here is to consider such large sample theory for the maximum distance to the origin, and the related maximum “interpoint distance,” in multidimensions. We show that for a family of spherically symmetric distributions, these statistics have a Gumbel-type limit, generalizing several existing results. We also discuss the other two types of limit laws and suggest some open problems. This work complements our earlier study on the minimum interpoint distance.

We consider whether ergodic Markov chains with bounded step size remain bounded in probability when their transitions are modified by an adversary on a bounded subset. We provide counterexamples to show that the answer is no in general, and prove theorems to show that the answer is yes under various additional assumptions. We then use our results to prove convergence of various adaptive Markov chain Monte Carlo algorithms.

The auto-cross covariance matrix is defined as

\[\mathbf{M}_{n}=\frac{1}{2T}\sum_{j=1}^{T}(\mathbf{e}_{j}\mathbf{e}_{j+\tau}^{*}+\mathbf{e}_{j+\tau}\mathbf{e}_{j}^{*}),\] where $\mathbf{e}_{j}$’s are $n$-dimensional vectors of independent standard complex components with a common mean 0, variance $\sigma^{2}$, and uniformly bounded $2+\eta$th moments and $\tau$ is the lag. Jin et al. [Ann. Appl. Probab. 24 (2014) 1199–1225] has proved that the LSD of $\mathbf{M}_{n}$ exists uniquely and nonrandomly, and independent of $\tau$ for all $\tau\ge1$. And in addition they gave an analytic expression of the LSD. As a continuation of Jin et al. [Ann. Appl. Probab. 24 (2014) 1199–1225], this paper proved that under the condition of uniformly bounded fourth moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of $\mathbf{M}_{n}$ for all large $n$. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of $\mathbf{M}_{n}$ are also obtained.