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223
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
"... This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Bor ..."
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Cited by 25 (6 self)
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This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Borel if it is a Borel subset of X². Given two
A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN
, 1998
"... The aim of this paper is to provide a viable measuretheoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors ..."
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Cited by 20 (10 self)
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The aim of this paper is to provide a viable measuretheoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus constitute the `right' class for general stochastic modeling. The primary concern of the paper is to characterize those hyperfinite processes satisfying the exact law of large numbers by using the basic notions of conditional expectation, orthogonality, uncorrelatedness and independence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed via a biorthogonal expansion of the KarhunenLoeve type and via the representation in terms of the simpler hyperfinite Loeb counting spaces. A universality property for atomless Loeb product spaces is formulated to show the abun...
Ambiguity without a State Space
, 2003
"... Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyzes a third domain of preference: sets of consequential lotte ..."
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Cited by 18 (1 self)
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Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyzes a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of Knightian ambiguity without explicitly invoking any state space. We characterize a representation that integrates a monotone transformation of first order expected utility with respect to a second order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion and uniquely characterize absolute ambiguity neutrality. Finally, we discuss applications of the theory: reinsurance, games, and a mean–variance–ambiguity portfolio frontier.
Reputational Cheap Talk
 Mimeo, London Business School
, 2004
"... This paper analyzes information reporting by a privately informed expert concerned about being perceived to have accurate information. When the expert’s reputation is updated on the basis of the report as well as the realized state, the expert typically does not wish to truthfully reveal the signal ..."
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Cited by 18 (2 self)
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This paper analyzes information reporting by a privately informed expert concerned about being perceived to have accurate information. When the expert’s reputation is updated on the basis of the report as well as the realized state, the expert typically does not wish to truthfully reveal the signal observed. The incentives to deviate from truthtelling are characterized and shown to depend on the information structure. In equilibrium, experts can credibly communicate only part of their information. Our results also hold when experts have private information about their own accuracy and care about their reputation relative to others.
Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems
 Ann. Appl. Prob
, 1996
"... In the paper we study interacting particle approximations of discrete time and measure valued dynamical systems. Such systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the socalled particle density profiles to converge to the desired ..."
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Cited by 17 (6 self)
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In the paper we study interacting particle approximations of discrete time and measure valued dynamical systems. Such systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the socalled particle density profiles to converge to the desired distribution when the number of particles is growing. The strength of our approach is that is applicable to a large class of measure valued dynamical system arising in engineering and particularly in nonlinear filtering problems. Our second objective is to use these results to solve numerically the nonlinear filtering equation. Examples arising in fluid mechanics are also given. 1 Introduction 1.1 Measure valued processes Let (E; fi(E)) be a locally compact and separable metric space, endowed with a Borel oefield, state space. Denote by P(E) be the space of all probability measures on E with the weak topology. The aim of this work is the design of a stochastic particle system approach fo...
Law of large number limits for manyserver queues
, 2007
"... Abstract. This work considers a manyserver queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well a ..."
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Cited by 17 (2 self)
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Abstract. This work considers a manyserver queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well as a measurevalued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, in the timehomogeneous setting, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these
The Complex Moment Problem and Subnormality: A Polar Decomposition Approach
, 1998
"... . It has been known that positive definiteness does not guarantee for a bisequence to be a complex moment one. However, it turns out that positive definite extendibility does (Theorems 1 and 22), and this is the main theme of this paper. The main tool is, generally understood, polar decomposition. S ..."
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Cited by 16 (4 self)
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. It has been known that positive definiteness does not guarantee for a bisequence to be a complex moment one. However, it turns out that positive definite extendibility does (Theorems 1 and 22), and this is the main theme of this paper. The main tool is, generally understood, polar decomposition. So as to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context (Theorems 9 and 29). All this enables us to prove: characterizations of subnormality of unbounded operators having invariant domain (Theorems 37 and 39) and their further applications (Theorems 41 and 43), and description of complex moment problem on real algebraic curves (Theorems 52 and 56). The latter question was completed in Appendix in which we relate the complex moment problem to the twodimensional real one, with emphasis on real algebraic sets. 1991 Mathematics Subject Classification. Primary 44A60, 47B20, 43A35; Secondary 60B99. Key words and phrase...
Heavytraffic limits for waiting times in manyserver queues with abandonments
, 2008
"... In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stop ..."
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Cited by 15 (8 self)
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In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stopped arrival processes as in the proof of Theorem 6.3.
Bootstrapping Autoregressive Processes with Possible Unit Roots
, 1999
"... An important question in applied work is how to bootstrap autoregressive processes involving highly persistent time series of unknown order of integration. In this paper, we show that in many cases of interest in applied work the standard bootstrap algorithm for unrestricted autoregressions remains ..."
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Cited by 15 (1 self)
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An important question in applied work is how to bootstrap autoregressive processes involving highly persistent time series of unknown order of integration. In this paper, we show that in many cases of interest in applied work the standard bootstrap algorithm for unrestricted autoregressions remains valid for processes with exact unit roots � no pretests are required, at least asymptotically, and applied researchers may proceed as in the stationary case. Specifically, we prove the firstorder asymptotic validity of bootstrapping any linear combination of the slope parameters in autoregressive models with drift. We also establish the bootstrap validity for the marginal distribution of slope parameters and for most linear combinations of slope parameters in higherorder autoregressions without drift. The latter result is in sharp contrast to the wellknown bootstrap invalidity result for the random walk without drift. A simulation study examines the finitesample accuracy of the bootstrap approximation both for integrated and for nearintegrated processes. We find that in many, but not all circumstances, the bootstrap distribution closely approximates the exact finitesample distribution.
Lectures on Young Measure Theory and its Applications in Economics
 Rend. Istit. Mat. Univ. Trieste
, 1998
"... this paper we work with the following hypothesis: ..."