We consider regular variation of a Lévy process X := (Xt)t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

We study a class of tenable, irreducible, nondegenerate zero-balanced Pólya urn schemes. We give a full characterization of the class by sufficient and necessary conditions. Only forms with a certain cyclic structure in their replacement matrix are admissible. The scheme has a steady state into proportions governed by the principal (left) eigenvector of the average replacement matrix. We study the gradual change for any such urn containing n → ∞ balls from the initial condition to the steady state. We look at the status of an urn starting with an asymptotically positive proportion of each color after jn draws. We identify three phases of jn: the growing sublinear, the linear, and the superlinear. In the growing sublinear phase the number of balls of different colors has an asymptotic joint multivariate normal distribution, with mean and covariance structure that are influenced by the initial conditions. In the linear phase a different multivariate normal distribution kicks in, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. We give interpretations for how the results in different phases conjoin at the ‘seam lines’. In fact, these Gaussian phases are all manifestations of one master theorem. The results are obtained via multivariate martingale theory. We conclude with some illustrating examples.

Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation, such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. In this paper, models with finitely-many alleles are considered, and an urn construction related to the coalescent is used to derive approximate closed-form sampling formulae for an arbitrary irreducible recurrent mutation model or for a reversible recurrent mutation model, depending on whether the number of distinct observed allele types is at most three or four, respectively. It is demonstrated empirically that the formulae derived here are highly accurate when the per-base mutation rate is low, which holds for many biological organisms.

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with remarkable accuracy.

We consider a model of fixed size N = 2l in which there are l generations of daughter cells and a stem cell. In each generation i there are 2i−1 daughter cells. At each integral time unit the cells split so that the stem cell splits into a stem cell and generation 1 daughter cell and the generation i daughter cells become two cells of generation i+1. The last generation is removed from the population. A stem cell acquires first and second mutations at rates u1 and u2, and a daughter cell acquires first and second mutations at rates v1 and v2. We find the distribution for the time it takes to acquire two mutations as N goes to ∞ and the mutation rates go to 0. The mutation rates may tend to 0 at different speeds. We also find the distribution for the locations of the mutations. In particular, we determine whether or not the mutations occur on a stem cell and if not, at what generation in the daughter cells they occur. Several outcomes are possible, depending on how fast the rates go to 0. The model considered has been proposed by Komarova (2007) as a model for colon cancer.

A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

The ‘coupon collection problem’ refers to a class of occupancy problems in which j identical items are distributed, independently and at random, to n cells, with no restrictions on multiple occupancy. Identifying the cells as coupons, a coupon is ‘collected’ if the cell is occupied by one or more of the distributed items; thus, some coupons may never be collected, whereas others may be collected once or twice or more. We call the number of coupons collected exactly r times coupons of type r. The coupon collection model we consider is general, in that a random number of purchases occurs at each stage of collecting a large number of coupons; the sample sizes at each stage are independent and identically distributed according to a sampling distribution. The joint behavior of the various types is an intricate problem. In fact, there is a variety of joint central limit theorems (and other limit laws) that arise according to the interrelation between the mean, variance, and range of the sampling distribution, and of course the phase (how far we are in the collection processes). According to an appropriate combination of the mean of the sampling distribution and the number of available coupons, the phase is sublinear, linear, or superlinear. In the sublinear phase, the normalization that produces a Gaussian limit law for uncollected coupons can be used to obtain a multivariate central limit law for at most two other types — depending on the rates of growth of the mean and variance of the sampling distribution, we may have a joint central limit theorem between types 0 and 1, or between types 0, 1, and 2. In the linear phase we have a multivariate central limit theorem among the types 0, 1,…, k for any fixed k.

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let {Xn, n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component Sn takes values on the real line , and is adjoined to the chain such that {Sn, n ≥ 1} is a Markov random walk. Assume that Sn = ∑k=1nξk, and that {ξn, n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of Sn / √n, we prove a self-normalized central limit theorem for Sn under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

We develop techniques for computing the asymptotics of the first and second moments of the number TN of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[TN] of TN (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of TN (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L2 of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

This paper deals with discrete-time Markov decision processes (MDPs) under constraints where all the objectives have the same form of expected total cost over the infinite time horizon. The existence of an optimal control policy is discussed by using the convex analytic approach. We work under the assumptions that the state and action spaces are general Borel spaces, and that the model is nonnegative, semicontinuous, and there exists an admissible solution with finite cost for the associated linear program. It is worth noting that, in contrast to the classical results in the literature, our hypotheses do not require the MDP to be transient or absorbing. Our first result ensures the existence of an optimal solution to the linear program given by an occupation measure of the process generated by a randomized stationary policy. Moreover, it is shown that this randomized stationary policy provides an optimal solution to this Markov control problem. As a consequence, these results imply that the set of randomized stationary policies is a sufficient set for this optimal control problem. Finally, our last main result states that all optimal solutions of the linear program coincide on a special set with an optimal occupation measure generated by a randomized stationary policy. Several examples are presented to illustrate some theoretical issues and the possible applications of the results developed in the paper.

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motion B. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup0≤s≤tB(s) < U} for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to be homothetic: all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.

Let {Xi} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C1, C2,…) be a nonnegative random vector independent of the {Xi} with N∈ℕ∪ {∞}. We study the weighted random sum SN=∑{i=1}NCiXi, and its maximum, MN=sup{1≤kN+1∑i=1kCiXi. This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄(x/Ci)] as x→∞. When E[X1]>0 and the distribution of ZN=∑ i=1NCi is also intermediate regularly varying, we obtain the asymptotics P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄}(x/Ci)] +P(ZN > x/E[X1]). For completeness, when the distribution of ZN is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(MN > x) ∼ P(SN > x)∼ P(ZN > x / E[X1] hold.

Let {Xt, t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θt, t ≥ 1} be a sequence of positive random variables independent of the sequence {Xt, t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X(∞) = ∑t=1∞ΘtXt+ (where X+ = max{0, X}) and max1≤k<∞∑t=1kΘtXt, and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X1, X2,… as independent and identically distributed positive random variables. If X1 has a regularly varying tail of index -α, where α > 0, and if {Θt, t ≥ 1} is a positive sequence of random variables independent of {Xt}, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θt, t ≥ 1}, X(∞) = ∑t=1∞ΘtXt converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X(∞) has a regularly varying tail then does X1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑t=1∞Θt along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.