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478
Metrics for Labelled Markov Systems
, 2001
"... The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of ..."
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Cited by 49 (10 self)
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The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of probabilistic processes. In a situation where the process behaviour has a quantitative aspect there should be a more robust approach to process equivalence. This paper studies a metric between labelled Markov processes. This metric has the property that processes are at zero distance if and only if they are bisimilar. The metric is inspired by earlier work on logics for characterizing bisimulation and is related, in spirit, to the Hutchinson metric.
Laws of Large Numbers for Dynamical Systems with Randomly Matched Individuals
 Journal of Economic Theory
, 1992
"... Biologists and economists have analyzed populations where each individual interacts with randomly selected individuals. The random matching generates a very complicated stochastic system. Consequently biologists and economists have approximated such a system with a deterministic system. The justitic ..."
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Cited by 46 (0 self)
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Biologists and economists have analyzed populations where each individual interacts with randomly selected individuals. The random matching generates a very complicated stochastic system. Consequently biologists and economists have approximated such a system with a deterministic system. The justitication for such an approximation is that the population is assumed to be very large and thus some law of large numbers must hold. This paper gives a characterization of random matching schemes for countably infinite populations. In particular this paper shows that there exists a random matching scheme such that the stochastic system and the deterministic system are the same. Journal of Economic Literature Classification
Maximum Likelihood Estimation of a Binary Choice Model with Random Coe¢ cients of Unknown Distribution
 Journal of Econometrics
, 1998
"... We consider a binary response model y = 1 { x/ ~ + £ ~ 0} with x 1 1 1 1 1 independent.of the unobservables (~l'£l) ' Nq finitedimensional parametric restrictions are imposed on F, the Joint distribution of ( ~,£). A 011 nonparametric maximum likelihood estimator for F is shown to be con ..."
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Cited by 46 (0 self)
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We consider a binary response model y = 1 { x/ ~ + £ ~ 0} with x 1 1 1 1 1 independent.of the unobservables (~l'£l) ' Nq finitedimensional parametric restrictions are imposed on F, the Joint distribution of ( ~,£). A 011 nonparametric maximum likelihood estimator for F is shown to be consistent. o We analyze some conditions under which F is or is not identified. We find o that certain moments of Fo are not identified, even when the model is normalized by fixing one variance. The correlation matrix of (~l'£l) is not identified. We also provide some Monte Carlo evidence on the small sample performance of our estimator. 1.
RealTime Queues in Heavy Traffic with EarliestDeadlineFirst Queue Discipline
, 2000
"... This paper introduces a new aspect of queueing theory, the study of systems that service customers with specic timing requirements (e.g. due dates or deadlines). Unlike standard queueing theory in which common performance measures are customer delay, queue length and server utilization, realtime qu ..."
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Cited by 44 (5 self)
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This paper introduces a new aspect of queueing theory, the study of systems that service customers with specic timing requirements (e.g. due dates or deadlines). Unlike standard queueing theory in which common performance measures are customer delay, queue length and server utilization, realtime queueing theory focuses on the ability of a queue discipline to meet customer timing requirements, e.g., the fraction of customers who meet their requirements and the distribution of customer lateness. It also focuses on queue control policies to reduce or minimize lateness, although these control aspects are not explicitly addressed in this paper. To study these measures, one must keep track of the leadtimes (deadline minus current time) of each customer, hence the system state is of unbounded dimension. A heavy trac analysis is presented for the earliest deadline rst (EDF) scheduling policy. This analysis decomposes the behavior of the realtime queue into two parts: the number in the sys...
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 42 (7 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
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 41 (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.
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 35 (4 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
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 33 (14 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...
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 32 (4 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.