| The relationship is studied between possibility and necessity measures dened on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural extensions. This leads to interesting alternative formulas for the calculation of these natural extensions. Keywords|Upper probability, upper prevision, coherence, natural extension, possibility measure, random sets. I. Introduction P OSSIBILITY measures were introduced by Zadeh [1] in 1978. In his view, these supremum preserving set functions are a mathematical representation of the information conveyed by typical armative statements in natural language. For recent discussions of this interpretation within the behavioural framework of the theory of imprecise probabilities, we refer to [2], [3], [4]. Supremum preserving set functions can also be found in the literature under a number o...

In a recent paper Baier, Haverkort, Hermanns and Katoen [BHHK00], analyzed a new way of model-checking formulas of a logic for continuoustime processes - called Continuous Stochastic Logic (henceforth CSL) { against continuous-time Markov chains { henceforth CTMCs. One of the important results of that paper was the proof that if two CTMCs were bisimilar then they would satisfy exactly the same formulas of CSL. This raises the converse question { does satisfaction of the same collection of CSL formulas imply bisimilarity? In other words, given two CTMCs which are known to satisfy exactly the same formulas of CSL does it have to be the case that they are bisimilar? We prove that the answer to the question just raised is \yes". In fact we prove a signi cant extension, namely that a subset of CSL suces even for systems where the state-space may be a continuum. Along the way we prove a result to the eect that the set of Zeno paths has measure zero provided that the transition rates are bounded.

Abstract. We show that every real nonnegative polynomial f can be approximated as closely as desired (in the l1-norm of its coefficient vector) by a sequence of polynomials {fɛ} that are sums of squares. The novelty is that each fɛ has a simple and explicit form in terms of f and ɛ. Key words. Real algebraic geometry; positive polynomials; sum of squares; semidefinite programming. AMS subject classifications. 12E05, 12Y05, 90C22 1. Introduction. The

The subjects of this thesis are unsupervised learning in general, and principal curves in particular. Principal curves were originally defined by Hastie \cite{Has84} and Hastie and Stuetzle \cite{HaSt89} (hereafter HS) to formally capture the notion of a smooth curve passing through the ``middle'' of a $d$-dimensional probability distribution or data cloud. Based on the definition, HS also developed an algorithm for constructing principal curves of distributions and data sets. The field has been very active since Hastie and Stuetzle's groundbreaking work. Numerous alternative definitions and methods for estimating principal curves have been proposed, and principal curves were further analyzed and compared with other unsupervised learning techniques. Several applications in various areas including image analysis, feature extraction, and speech processing demonstrated that principal curves are not only of theoretical interest, but they also have a legitimate place in the family of practical unsupervised learning techniques. Although the concept of principal curves as considered by HS has several appealing characteristics, complete theoretical analysis of the model seems to be rather hard. This motivated us to redefine principal curves in a manner that allowed us to carry out extensive theoretical analysis while preserving the informal notion of principal curves. Our first contribution to the area is, hence, a new {\em theoretical model} that is analyzed by using tools of statistical learning theory. Our main result here is the first known consistency proof of a principal curve estimation scheme. The theoretical model proved to be too restrictive to be practical. However, it inspired the design of a new {\em practical algorithm} to estimate principal curves based on data. The polygonal line algorithm, which compares favorably with previous methods both in terms of performance and computational complexity, is our second contribution to the area of principal curves. To complete the picture, in the last part of the thesis we consider an {\em application} of the polygonal line algorithm to hand-written character skeletonization.

High-resolution bounds in lossy coding of a real memoryless source are considered when side information is present. Let be a "smooth" source and let be the side information. First we treat the case when both the encoder and the decoder have access to and we establish an asymptotically tight (high-resolution) formula for the conditional rate-distortion function ( ) for a class of locally quadratic distortion measures which may be functions of the side information. We then consider the case when only the decoder has access to the side information (i.e., the "Wyner--Ziv problem"). For side-information-dependent distortion measures, we give an explicit formula which tightly approximates the Wyner--Ziv rate-distortion function ( ) for small under some assumptions on the joint distribution of and . These results demonstrate that for side-information-dependent distortion measures the rate loss ( ) ( ) can be bounded away from zero in the limit of small . This contrasts the case of distortion measures which do not depend on the side information where the rate loss vanishes as 0.

this paper we work with the following hypothesis:

The employment of ‘Strong Laws of Large Numbers ’ is instrumental to the analysis of system estimation and identification strategies. However, the vast bulk of such laws, as presented in the wider literature, assume independence or at least uncorrelatedness of random components and these assumptions are quite restrictive from an engineering point of view. By way of contrast, this paper shows how to establish strong laws for possibly non-stationary random processes with very general dependence structure. Brief examples are provided that illustrate the utility of the Strong Law of Large Numbers presented.

Optimal scalar quantization subject to an entropy-constraint is studied. First the problem of nding analytically an optimal entropy-constrained scalar quantizer (ECSQ) is considered. For a wide class of dierence distortion measures including rth power distortions with r > 0, it is proved that if the source is uniformly distributed over an interval, then for any entropy constraint R (in bits), an optimal quantizer has N = 2 R interval cells such that N 1 cells have equal length d and one cell has length c d. Based on this result, a parametric representation of the minimum achievable distortion D h (R) as a function of the entropy constraint R is obtained for a uniform source. Contrary to earlier expectations, the D h (R) curve turns out to be nonconvex in general. In particular, for the squared error distortion it is shown that D h (R) is a piecewise concave function. The structural properties of optimal ECSQs for more general source distributions are also investigated. In...

New fundamentals of Young measure convergence

The paper shows how to convert statistical prediction sets into worst case hedging strategies for derivative securities. The prediction sets can, in particular, be ones for volatilities and correlations of the underlying securities, and for interest rates. This permits a transfer of statistical conclusions into prices for options and similar financial instruments. A prime feature of our results is that one can construct the trading strategy as if the prediction set had a 100 % probability. If, in fact, the set has probability 1−α, the hedging strategy will work with at least the same probability. Different types of prediction regions are considered. The starting value A0 for the trading strategy corresponding to the 1 − α prediction region is a form of long term value at risk. At the same time, A0 is coherent.