A Modern Approach to Probability Theory

Fristedt, Bert; Gray, Lawrence

doi:10.1007/978-1-4899-2837-5

Cited by 286 publications

(216 citation statements)

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“…In words, this means that the distribution of X k becomes close to that of X when k is large. For more on this, other equivalent definitions, and extensive coverage of convergence in distribution, see Billingsley (1968), Fristedt and Gray (1997), or other probability texts. As mentioned in the Introduction, our results follow from the assumption that the vector of the normalized number of arrivals converges in distribution to the vector of the original expected number of arrivals:…”

Section: Asymptotic Resultsmentioning

confidence: 99%

Asymptotic Behavior of an Allocation Policy for Revenue Management

Cooper

2002

Operations Research

143

124

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Revenue management has become an important tool in the airline, hotel, and rental car industries. We describe asymptotic properties of revenue management policies derived from the solution of a deterministic optimization problem. Our primary results state that, within a stochastic and dynamic framework, solutions arising out of a single well-known linear program can be used to generate allocation policies for which the normalized revenue converges in distribution to a constant upper bound on the optimal value. We also show similar asymptotic results for expected revenues. In addition, we describe counterintuitive behavior that can occur when allocations are updated during the booking process (updating allocations can lead to lower expected revenue). These results add to the understanding of allocation policies and help to make concrete the statement that simple policies from easy-to-solve formulations can be relatively effective, even when analyzed in the more realistic stochastic and dynamic framework.

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Section: Asymptotic Resultsmentioning

confidence: 99%

Asymptotic Behavior of an Allocation Policy for Revenue Management

Cooper

2002

Operations Research

143

124

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“…X closed, be a Polish subspace, i.e., X is separable (it has a countable dense subset) and complete (every Cauchy sequence in X converges to a point x ∈ X) [11,12]. We denote as AC(X) the collection of all probability measures µ on (X, B(X)) which are absolutely continuous with respect to (w.r.t.)…”

Section: Definitionsmentioning

confidence: 99%

On convergence properties of Shannon entropy

Piera

Parada

2009

Probl Inf Transm

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Convergence properties of Shannon Entropy are studied. In the differential setting, it is shown that weak convergence of probability measures, or convergence in distribution, is not enough for convergence of the associated differential entropies. A general result for the desired differential entropy convergence is provided, taking into account both compactly and uncompactly supported densities. Convergence of differential entropy is also characterized in terms of the Kullback-Liebler discriminant for densities with fairly general supports, and it is shown that convergence in variation of probability measures guarantees such convergence under an appropriate boundedness condition on the densities involved. Results for the discrete setting are also provided, allowing for infinitely supported probability measures, by taking advantage of the equivalence between weak convergence and convergence in variation in this setting.

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“…It is known (see [16,17,20,26,33]) that convergence P n → P in the Prohorov metric is equivalent to the statement…”

Section: Theoretical Framework For General Delay Systemmentioning

confidence: 99%

Inverse problems for a class of measure dependent dynamical systems

Banks¹,

Bortz²

2005

Journal of Inverse and Ill-posed Problems

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ABSTRACT. We consider a class of probability measure dependent dynamical systems which arise in the study of multiscale phenomena in diverse fields such as immunological population dynamics, viscoelasticity of polymers and rubber, and polarization in dielectric materials. We develop an inverse problem framework for studying systems with distributed temporal delays. In particular, we establish conditions for existence and uniqueness of the forward problem and well-posedness (including method stability under numerical approximations) for the inverse problem of estimating the probability measures. We show that a motivating class of models of HIV infection dynamics satisfies all the conditions of our framework, thereby providing a theoretical foundation for inverse problem computations with these models.

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A Modern Approach to Probability Theory

Cited by 286 publications

References 0 publications

Asymptotic Behavior of an Allocation Policy for Revenue Management

Asymptotic Behavior of an Allocation Policy for Revenue Management

On convergence properties of Shannon entropy

Inverse problems for a class of measure dependent dynamical systems

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