We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variable S(t), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at time t do exist, and satisfy the PDEs.

A theorem that gives the optimal control of Gaussian processes using the mathematical expectation of a function of the time and the place where the uncontrolled processes hit the boundary of the stopping region for the first time is proved. The result obtained in this note is an extension of a theorem due to Whittle.

Consider a random walk S=(Sn: n≥0) that is ‘perturbed’ by a stationary sequence (ξn: n≥0) to produce the process S=(Sn+ξn: n≥0). In this paper, we are concerned with developing limit theorems and approximations for the distribution of Mn=max{Sk+ξk: 0≤k≤n} when the random walk has a drift close to 0. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walks and extreme-value limit theory.

Consider a processor-sharing queue placed in a quasi-reversible network in equilibrium. This note explains why the expected sojourn time of a customer in such a queue is proportional to his service time.

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

We consider stochastic replicator processes for games that are composed of finitely many trials. Several general results on the relation between Nash equilibria and the long-run behaviour of the stochastic processes are proved. In particular, a sufficient condition is given for almost sure convergence to a state where everyone plays in every trial a strict Nash equilibrium. The results are applied to multiple-trial conflicts based on wars of attrition and on sperm competition games with fair raffles, respectively.

We show that Niu&s (1980) conjecture is true for Poisson arrivals and light traffic. We also show that a weak version of the conjecture is true for a special case in heavy traffic.

We perform an analysis of the potential time inhomogeneity in the dependence between multiple financial time series. To this end, we use the framework of copula theory and tackle the question of whether dependencies in such a case can be assumed constant throughout time or rather have to be modeled in a time-inhomogeneous way. We focus on parametric copula models and suitable inference techniques in the context of a special copula-based multivariate time series model. A recent result due to Chan et al. (2009) is used to derive the joint limiting distribution of local maximum-likelihood estimators on overlapping samples. By restricting the overlap to be fixed, we establish the limiting law of the maximum of the estimator series. Based on the limiting distributions, we develop statistical homogeneity tests, and investigate their local power properties. A Monte Carlo simulation study demonstrates that bootstrapped variance estimates are needed in finite samples. Empirical analyses on real-world financial data finally confirm that time-varying parameters are an exception rather than the rule.

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for

Ln(s) is asymptotically Poisson.

Let ℬs([a,b];μ1,μ2,…,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,…,μs-1 as their first s-1 moments. In this paper we examine some aspects of the structure of ℬs([a,b];μ1,μ2,…,μs-1) and of the s-convex stochastic extrema in it. Using representation results of moment matrices à la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in ℬs([a,b];μ1,μ2,…,μs-1) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in ℬs([a,b];μ1,μ2,…,μs-1) by one of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.

Starting with a stationary point process on the line with points one unit apart, simultaneously replace each point by a point located uniformly between the original point and its right-hand neighbor. Iterating this transformation, we obtain convergence to a limiting point process, which we are able to identify. The example of the uniform distribution is for purposes of illustration only; in fact, convergence is obtained for almost any distribution on [0, 1]. In the more general setting, we prove the limiting distribution is invariant under the above transformation, and that for each such transformation, a large class of initial processes leads to the same invariant distribution. We also examine the covariance of the limiting sequence of interval lengths. Finally, we identify those invariant distributions with independent interval lengths, and the transformations from which they arise.

The stochastic control problem of tracking a Brownian motion by a process of bounded variation is reduced to a control problem with reflection at the origin, and the latter is related to a question of optimal stopping of Brownian motion absorbed at the origin. Direct probabilistic arguments can be used to show equivalences between the various problems.

In this paper we study the so-called random coeffiecient autoregressive models (RCA models) and (generalized) autoregressive models with conditional heteroscedasticity (ARCH/GARCH models). Both models can be represented as random systems with complete connections. Within this framework we are led (under certain conditions) to CL-regular Markov processes and we will give conditions under which (i) asymptotic stationarity, (ii) a law of large numbers and (iii) a central limit theorem can be shown for the corresponding models.