We prove that proper coloring distinguishes between block factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and asked whether a $k$ -dependent $q$ -coloring exists for any $k$ and $q$ . We give a complete answer by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovász local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving $d$ dimensions and shifts of finite type; in fact, any nondegenerate shift of finite type also distinguishes between block factors and finitely dependent processes.

The aim of this paper is to find a formula for the double Laplace transform of the truncated variation of a Brownian motion with drift. In order to find the double Laplace transform, we also prove some identities for the Brownian motion with drift, which may be of independent interest.

The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n≥1, choose a ball from the urn uniformly at random. With probability ½<p<1, return the ball to the urn along with another ball of the same colour. With probability 1−p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri (2015), (2016) and Thörnblad (2015). We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1−p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p=1).

The best-choice problem and the duration problem, known as versions of the secretary problem, are concerned with choosing an object from those that appear sequentially. Let (B,p) denote the best-choice problem and (D,p) the duration problem when the total number N of objects is a bounded random variable with prior p=(p1, p2,...,pn) for a known upper bound n. Gnedin (2005) discovered the correspondence relation between these two quite different optimal stopping problems. That is, for any given prior p, there exists another prior q such that (D,p) is equivalent to (B,q). In this paper, motivated by his discovery, we attempt to find the alternate correspondence {p(m),m≥0}, i.e. an infinite sequence of priors such that (D,p(m-1)) is equivalent to (B,p(m)) for all m≥1, starting with p(0)=(0,...,0,1). To be more precise, the duration problem is distinguished into (D1,p) or (D2,p), referred to as model 1 or model 2, depending on whether the planning horizon is N or n. The aforementioned problem is model 1. For model 2 as well, we can find the similar alternate correspondence {p[m],m≥ 0}. We treat both the no-information model and the full-information model and examine the limiting behaviors of their optimal rules and optimal values related to the alternate correspondences as n→∞. A generalization of the no-information model is given. It is worth mentioning that the alternate correspondences for model 1 and model 2 are respectively related to the urn sampling models without replacement and with replacement.

We consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.

We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color, given the past, is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.

We obtain a complete description of anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian, and ‘intermediate’ infinitely divisible random fields. The asymptotic form of the covariance function of the random grain model is obtained. Application to superimposed network traffic is included.

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy-tailed steps, the limiting behavior of extreme observations until a given time t tends to be rather involved. We describe the asymptotics and extend several partial results which appeared in this setting. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous-time random walks.

The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log1∕p n and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

In this paper we study a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward, and signed (positive and negative) switching costs. Using optimal stopping theory for discrete-parameter stochastic processes, we extend a well-known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs.

When a random field (Xt,t∈ℝ2) is thresholded on a given level u, the excursion set is given by its indicator 1[u, ∞)(Xt). The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets as, e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular, Rice methods, and from integral and stochastic geometry.

We study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

We study G/G/∞ queues with renewal alternating service interruptions, where the service station experiences `up' and `down' periods. The system operates normally in the up periods, and all servers stop functioning while customers continue entering the system during the down periods. The amount of service a customer has received when an interruption occurs will be conserved and the service will resume when the down period ends. We use a two-parameter process to describe the system dynamics: Xr(t,y) tracking the number of customers in the system at time t that have residual service times strictly greater than y. The service times are assumed to satisfy either of the two conditions: they are independent and identically distributed with a distribution of a finite support, or are a stationary and weakly dependent sequence satisfying the ϕ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavy-traffic asymptotic regime where the arrival rate gets large and service time distribution is fixed, and the interruption down times are asymptotically negligible while the up times are of the same order as the service times. We show the functional law of large numbers and functional central limit theorem (FCLT) for the process Xr(t,y) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻, L1)) endowed with the Skorokhod M1 topology. The limit processes in the FCLT possess a stochastic decomposition property.

We study partial hedging for game options in markets with transaction costs bounded from below. More precisely, we assume that the investor's transaction costs for each trade are the maximum between proportional transaction costs and a fixed transaction cost. We prove that in the continuous-time Black‒Scholes (BS) model, there exists a trading strategy which minimizes the shortfall risk. Furthermore, we use binomial models in order to provide numerical schemes for the calculation of the shortfall risk and the corresponding optimal portfolio in the BS model.

In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.

We review, implement, and compare numerical integration schemes for spatially bounded diffusions stopped at the boundary which possess a convergence rate of the discretization error with respect to the timestep h higher than . We address specific implementation issues of the most general-purpose of such schemes. They have been coded into a single Matlab program and compared, according to their accuracy and computational cost, on a wide range of problems in up to ℝ48. The paper is self-contained and the code will be made freely downloadable.

We study the exciton diffusion in organic semiconductors from a macroscopic viewpoint. In a unified way, we conduct the equivalence analysis between Monte-Carlo method and diffusion equation model for photoluminescence quenching and photocurrent spectrum measurements, in both the presence and the absence of Förster energy transfer effect. Connections of these two models to Stern-Volmer method and exciton-exciton annihilation method are also specified for the photoluminescence quenching measurement.

Famously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially nonhomogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension d≥2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these elliptic random walks generalize the classical homogeneous Pearson‒Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.

Preferential attachment is a widely adopted paradigm for understanding the dynamics of social networks. Formal statistical inference, for instance GLM techniques, and model-verification methods will require knowing test statistics are asymptotically normal even though node- or count-based network data are nothing like classical data from independently replicated experiments. We therefore study asymptotic normality of degree counts for a sequence of growing simple undirected preferential attachment graphs. The methods of proof rely on identifying martingales and then exploiting the martingale central limit theorems.