The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

We show that under some slight assumptions, the positive sojourn time of a product of symmetric processes converges towards ½ as the number of processes increases. Monotony properties are then exhibited in the case of symmetric stable processes, and used, via a recurrence relation, to obtain upper and lower bounds on the moments of the occupation time (in the first and third quadrants) for two-dimensional Brownian motion. Explicit values are also given for the second and third moments in the n-dimensional Brownian case.

Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.

We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.

By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

Introduced recently in mathematical finance by Bayer et al. (2016), the rough Bergomi model has proved particularly efficient to calibrate option markets. We investigate some of its probabilistic properties, in particular proving a pathwise large deviations principle for a small-noise version of the model. The exponential function (continuous but superlinear) as well as the drift appearing in the volatility process fall beyond the scope of existing results, and a dedicated analysis is needed.

We examine a system of interacting random walks with leftward drift on ℤ, which begins with a single active particle at the origin and some distribution of inactive particles on the positive integers. Inactive particles become activated when landed on by other particles, and all particles beginning at the same point possess equal leftward drift. Once activated, the trajectories of distinct particles are independent. This system belongs to a broader class of problems involving interacting random walks on rooted graphs, referred to collectively as the frog model. Additional conditions that we impose on our model include that the number of frogs (i.e. particles) at positive integer points is a sequence of independent random variables which is increasing in terms of the standard stochastic order, and that the sequence of leftward drifts associated with frogs originating at these points is decreasing. Our results include sharp conditions with respect to the sequence of random variables and the sequence of drifts that determine whether the model is transient (meaning the probability infinitely many frogs return to the origin is 0) or nontransient. We consider several, more specific, versions of the model described, and a cleaner, more simplified set of sharp conditions will be established for each case.

We consider a supercritical branching process (Zn, n ≥ 0) with offspring distribution (pk, k ≥ 0) satisfying p0 = 0 and p1 > 0. By applying the self-normalized large deviation of Shao (1997) for independent and identically distributed random variables, we obtain the self-normalized large deviation for supercritical branching processes, which is the self-normalized version of the result obtained by Athreya (1994). The self-normalized large deviation can also be generalized to supercritical multitype branching processes.

The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

We consider the problem of optimal consumption of multiple goods in incomplete semimartingale markets. We formulate the dual problem and identify conditions that allow for the existence and uniqueness of the solution, and provide a characterization of the optimal consumption strategy in terms of the dual optimizer. We illustrate our results with examples in both complete and incomplete models. In particular, we construct closed-form solutions in some incomplete models.

For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability ϕ(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

In this paper we consider the factor analysis for Lévy-driven multivariate price models with stochastic volatility. Our main aim is to provide conditions on the volatility process under which we can possibly reduce the dimension of the driving Lévy motion. We find that these conditions depend on a particular form of the multivariate Lévy process. In some settings we concentrate on nondegenerate symmetric α-stable Lévy motions.

The selection model in population genetics is a dynamic system on the set of the probability distributions 𝒑=(p1,…,pn) of the alleles A1…,An, with pi(t+1) proportional to pi(t) multiplied by ∑jfi,jpj(t), and fi,j=fj,i interpreted as a fitness of the gene pair (Ai,Aj). It is known that 𝒑̂ is a locally stable equilibrium if and only if 𝒑̂ is a strict local maximum of the quadratic form 𝒑T𝒇𝒑. Usually, there are multiple local maxima and lim𝒑(t) depends on 𝒑(0). To address the question of a typical behavior of {𝒑(t)}, John Kingman considered the case when the fi,j are independent and [0,1]-uniform. He proved that with high probability (w.h.p.) no local maximum may have more than 2.49n1∕2 positive components, and reduced 2.49 to 2.14 for a nonbiological case of exponentials on [0,∞). We show that the constant 2.14 serves a broad class of smooth densities on [0,1] with the increasing hazard rate. As for a lower bound, we prove that w.h.p. for all k≤2n1∕3, there are many k-element subsets of [n] that pass a partial test to be a support of a local maximum. Still, it may well be that w.h.p. the actual supports are much smaller. In that direction, we prove that w.h.p. a support of a local maximum, which does not contain a support of a local equilibrium, is very unlikely to have size exceeding ⅔log2n and, for the uniform fitnesses, there are super-polynomially many potential supports free of local equilibriums of size close to ½log2n.

Let Xn(k) be the number of vertices at level k in a random recursive tree with n+1 vertices. We are interested in the asymptotic behavior of Xn(k) for intermediate levels k=kn satisfying kn→∞ and kn=o(logn) as n→∞. In particular, we prove weak convergence of finite-dimensional distributions for the process (Xn ([knu]))u>0, properly normalized and centered, as n→∞. The limit is a centered Gaussian process with covariance (u,v)↦(u+v)−1. One-dimensional distributional convergence of Xn(kn), properly normalized and centered, was obtained with the help of analytic tools by Fuchs et al. (2006). In contrast, our proofs, which are probabilistic in nature, exploit a connection of our model with certain Crump–Mode–Jagers branching processes.

We present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact simulation in this setting is challenging because of the presence of correlated local-time-like terms in the definition of RBM. We apply recently developed so-called ε-strong simulation techniques (also known as tolerance-enforced simulation) which allow us to provide a piecewise linear approximation to RBM with ε (deterministic) error in uniform norm. A novel conditional acceptance–rejection step is then used to eliminate the error. In particular, we condition on a suitably designed information structure so that a feasible proposal distribution can be applied.

Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

We study the conditions for unimodality of the lifetime distribution of a coherent system when the ordered component lifetimes in the system are described by generalized order statistics. Results for systems with independent and identically distributed lifetimes of components are included in this setting. The findings are illustrated with some examples for different types of systems. In particular, coherent systems with strictly bimodal density functions are presented in the case of independent standard uniform distributed lifetimes of components. Furthermore, we use the results to derive a sharp upper bound on the expected system lifetime in terms of the mean and the standard deviation of the underlying distribution.