Perturbation analysis of functionals of random measures

The present paper continues the work by Davidson, Krickeberg, Papangelou, and the author on proving, under weakest possible assumptions, that a stationary random measure η or a simple point process ξ on the space of k-flats in R d is a.s. invariant or a Cox process respectively. The problems for ξ and η are related by the fact that ξ is Cox whenever the Papangelou conditional intensity measure ζ of (a thinning of) ξ is a.s. invariant. In particular, η is shown to be a.s. invariant, whenever it is absolutely continuous with respect to some fixed measure μ and has no (so called) outer degeneracies. When k=d−2≧2, no absolute continuity is needed, provided that the first moments exist and that η has no inner degeneracies either. Under a certain regularity condition on ξ, it is further shown that ξ and ζ are simultaneously non-degenerate in either sense.

Stationary Point Processes and Random Measures

We consider stationary and ergodic tessellations X = Ξnn≥1 in Rd, where X is observed in a bounded and convex sampling window Wp ⊂ Rd. It is assumed that the cells Ξn of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J0 acting on Rd. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in Wp, as Wp ↑ Rd. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

We consider an M/G/∞ queue where the service station is subject to occasional interruptions which form an alternating renewal process ofup anddown periods. We show that under some natural conditions the random measure process associated with the residual service times of the customers is regenerative in the strict sense, and study its steady state characteristics. In particular we show that the steady state distribution of this random measure is a convolution of two distributions of (independent) random measures, one of which is associated with a standard M/G/∞ queue.

A class of second-order stationary random measures

Lévy processes time-changed by inverse subordinators have been intensively studied in the last years. Their importance in connection with non-local operators and semi-Markov dynamics is well understood, but, in our view, several questions remain open concerning the probabilistic structure of such processes. The time-changed Lévy processes are particularly useful to describe complex systems with fractional and/or anomalous dynamics. The purpose of our work is to analyze the features of the sample paths of such processes, focusing on a martingale-based approach. We introduce the fractional Poisson random measure as the main tool for dealing with the jump component of time-changed càdlàg processes. Further, the fractional random measure is an interesting and novel topic in itself, and thus, it is thoroughly analyzed in the paper. A central role in our analysis is then played by fractional Poisson integrals (involving the aforementioned fractional Poisson measure) which allow a useful description of the random jumps. We investigate these stochastic integrals and the martingale property of their compensated counterpart. Therefore, we are able to obtain a semimartingale representation of time-changed processes analogous to the celebrated Lévy–Itô decomposition. Finally, an approximation scheme of such random processes will be discussed.

We examine three of the principles of probabilistic potential theory in a nonclassical setting. These are: (i) the bounded maximum principle, (ii) the positive definiteness of the energy (of measures of bounded potential), and (iii) the condition that each semipolar set is polar. These principles are known to be equivalent in the context of two Markov processes in strong duality, when excessive functions are lower semicontinuous. We show that when the principles are appropriately formulated their equivalence persists in the wider context of a Borel right Markov processX with distinguished excessive measurem. We make no duality hypotheses andm need not be a reference measure. Our main tools are the stationary process (Y, Qm) associated withX andm, and a correspondence between potentials μU and certain random measures over (Y, Qm).

Continuous trading in an exchange economy under discontinuous dynamics: A resolution of the equity premium puzzle

By a classical result of Gray, Neuhoff and Shields (1975) the rhobar-distance between stationary processes is identified with an optimal stationary coupling problem of the corresponding stationary measures on the infinite product spaces. This is a modification of the optimal coupling problem from Monge--Kantorovich theory. In this paper we derive some general classes of examples of optimal stationary couplings which allow to calculate the rhobar distance in these cases in explicit form. We also extend the rhobar-distance to random fields and to general nonmetric distance functions and give a construction method for optimal stationary cbar-couplings. Our assumptions need in this case a geometric positive curvature condition.

Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on ℝd, derived from these processes using Hausdorff and projection measures are studied. The central limit theorem is formulated in a way which enables comparison of the various estimators of the intensity of the produced random measures. Approximate confidence intervals for the intensity are constructed. Their use is demonstrated in an example of length intensity estimation for the segment processes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper—which led to the awarding of the 1997 Nobel prize in Economic Sciences—Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its relative simplicity, the description of Brownian motion in advanced textbooks sometimes lacks an intuitive basis. The present exposition attempts to provide a simplified construction of standard Brownian motion based on a gambling analogy. This is followed by a description and explicit solution of two stochastic differential equations (known as arithmetic and geometric Brownian motion processes) that are driven by the standard Brownian motion process. The paper also illustrates the use of the Maple computer algebra system to simulate the standard and geometric Brownian motion processes.

Noise, chaos, and (ε, τ)-entropy per unit time

Point processes X of cylinders, compact sets (particles), or flats in ℝ d are mathematical models for fields of sets as they occur, e.g., in practical problems of image analysis and stereology. For the estimation of geometric quantities of such fields, mean value formulas for X are important. By a systematic approach, integral geometric formulas for curvature measures are transformed into density formulas for geometric point processes. In particular, a number of results which are known for stationary and isotropic Poisson processes of convex sets are generalized to nonisotropic processes, to non-Poissonian processes, and to processes of nonconvex sets. The integral geometric background (including recent results from translative integral geometry), the fundamentals of geometric point processes, and the resulting density formulas are presented in detail. Generalizations of the theory and applications in image analysis and stereology are mentioned shortly.

A method of both measurement of power parameters of a microwave wideband intense random signal and detection of a narrow-band signal with unknown frequency and power against a background of this wideband signal is presented. This method is based on nonheterodyne frequency and power conversion using a gyromagnetic converter that operates in two regimes: resonance detection and cross-multiplication. The wideband spectrum envelope of a random signal is visualized, and by switching two regimes and using correspondent filtration of the converted RF signal, the narrowband signals can be detected and measured. The block-scheme and operation of a two-channel measuring device combining both functions are discussed.

It was described the mathematical model of procedures for moving spectrally differentiating non-stationary random biomedical measurement signals. It is proved a possibility of additional diagnostic information to quantify the changes in electrical and mechanical activity of biological structures at the physical and chemical effects. It is shown an efficiency of using procedures developed for the analysis of activity chemically induced spontaneous contractions of the ureter and identification cardiodynamic violations in closed chest trauma, accompanied by heart injury