Persistent Random Walks in Stationary Environment

We study the behavior of persistent random walks (RW) on the integers in a random environment. A complete characterization of the almost sure limit behavior of these processes, including the law of large numbers, is obtained. This is done in a general situation where the environmental sequence of random variables is stationary and ergodic. Szász and Tóth obtained a central limit theorem when the ratio /, of right- and left-transpassing probabilities satisfies /a<1 a.s. (for a given constant a). We consider the case where / has wider fluctuations; we shall observe that an unusual situation arises: the RW may converge a.s. to infinity even with zero drift. Then, we obtain nonclassical limiting distributions for the RW. Proofs are based on the introduction of suitable branching processes in order to count the steps performed by the RW.     

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

On a Theorem of Dubins and Freedman

Under a notion of splitting the existence of a unique invariant probability, and a geometric rate of convergence to it in an appropriate metric, are established for Markov processes on a general state space S generated by iterations of i.i.d. maps on S. As corollaries we derive extensions of earlier results of Dubins and Freedman;⁽¹⁷⁾ Yahav;⁽³⁰⁾ and Bhattacharya and Lee⁽⁶⁾ for monotone maps. The general theorem applies in other contexts as well. It is also shown that the Dubins–Freedman result on the necessity of splitting in the case of increasing maps does not hold for decreasing maps, although the sufficiency part holds for both. In addition, the asymptotic stationarity of the process generated by i.i.d. nondecreasing maps is established without the requirement of continuity. Finally, the theory is applied to the random iteration of two (nonmonotone) quadratic maps each with two repelling fixed points and an attractive period-two orbit.     

Uniform dimension results of multi-parameter stable processes

The problem of uniform dimensions for multi-parameter processes, which may not possess the uniform stochastic Hölder condition, is investigated. The problem of uniform dimension for multi-parameter stable processes is solved. That is, ifZ is a stable (N,d, α)-process and αN ?d, then $\forall E \subseteq \mathbb{R}_ + ^N , \dim Z\left( E \right) = \alpha \cdot \dim E$ holds with probability 1, whereZ(E) = {x : ?t ∈E,Z _t =x} is the image set ofZ onE. The uniform upper bounds for multi-parameter processes with independent increments under general conditions are also given. Most conclusions about uniform dimension can be considered as special cases of our results.     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

Equivalence of a -functional with the approximation behavior of some linear means for abstract Fourier series

Within the setting of abstract Cesàro-bounded Fourier series a -functional is introduced and characterized by the convergence behavior of some linear means. Applications are given within the framework of Jacobi, Laguerre and Hermite expansions. In particular, Ditzian's (1996) equivalence result in the setting of Legendre expansions is covered.

     

Isothermal volume variations in lipid vesicle suspensions. A new evidence of intervesicle fusion kinetics

Summary In the present study of fusion between lipid vesicles performed by thermomechanical analysis, isothermal volume variation has been shown to be a reliable tool to follow these kinetics without introducing perturbing probes. In fact, the fusion process is accompanied by bilayer strain release which causes an overall volume decrease of the fused vesicles. Volumetric variations induced by side processes, such as adhesion or ion binding onto the vesicle surface, were accounted for in our measurements. Moreover, by the same technique we followed segregation effects of the membrane lipid components in mixed vesicles. The systems examined were neutral and anionic phospholipids containing vesicles. The role of temperature, vesicle size, lipid composition as well as the influence of different cations were also investigated.     

Maximum principle for control problems with uncertain horizon and variable discount rate

We consider an optimal control problem in which the time horizon is a random variable and the discount factor may depend on the past state and control values. This problem combines features of controlled piecewise deterministic processes and recursive utility maximization. Applying a simple transformation and a refined version of Halkin's proof of the maximum principle for optimal control problems on unbounded time intervals (Ref. 1), we obtain the maximum principle for the problem under consideration. Our assumptions are weaker than those of related results in the literature.This research was supported by a grant from the Austrian Science Foundation, Vienna, Austria.     

A characterization for mixtures of semi-Markov processes

Mixtures of recurrent semi-Markov processes are characterized through a partial exchangeability condition of the array of successor states and holding times. A stronger invariance condition on the joint law of successor states and holding times leads to mixtures of Markov laws.     

Quadratic assignment problems

This paper surveys quadratic assignment problems (QAP). At first several applications of this problem class are described and mathematical formulations of QAPs are given. Then some exact solution methods and good heuristics are outlined. Their computational behaviour is illustrated by numerical results. Further recent results on the asymptotic probabilistic behaviour of QAPs are outlined.