Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).
About: This article is published in The Mathematical Gazette.The article was published on 1995-11-01. It has received 2345 citations till now.
Citations
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TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
7,412 citations
TL;DR: The per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery, can be increased dramatically under this assumption, and a form of multiuser diversity via packet relaying is exploited.
Abstract: The capacity of ad hoc wireless networks is constrained by the mutual interference of concurrent transmissions between nodes. We study a model of an ad hoc network where n nodes communicate in random source-destination pairs. These nodes are assumed to be mobile. We examine the per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery. Under this assumption, the per-user throughput can increase dramatically when nodes are mobile rather than fixed. This improvement can be achieved by exploiting a form of multiuser diversity via packet relaying.
2,736 citations
Cites methods from "Stable non-Gaussian random processe..."
...2 There is no closed form for the distribution or density function of for general ; only the Laplace transform of its density is known explicitly [3], [ 9 ] and is given by . 3 This can be seen by comparing the Laplace transform of the density of nonnegative strictly stable random variables in [3, p. 448] with the expression for the characteristic function of general stable random variables in [9, p. 5]. Fig. 6. The normalized per-node ......
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...We have evaluated the asymptotic fraction of feasible pairs for the special case , because for this case, the normalized interference has a Levy distribution 2 [ 9 ], with cdf...
[...]
Book•
12 Jun 2007TL;DR: Random Fields and Geometry as discussed by the authors is a comprehensive survey of the general theory of Gaussian random fields with a focus on geometric problems arising in the study of random fields, including continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities.
Abstract: * Recasts topics in random fields by following a completely new way of handling both geometry and probability
* Significant exposition of the work of others in the field
* Presentation is clear and pedagogical
* Excellent reference work as well as excellent work for self study
This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined.
The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities.
"Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.
1,465 citations
22 Apr 2001
TL;DR: The per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery, can be increased dramatically when the nodes are mobile rather than fixed, by exploiting node mobility as a type of multiuser diversity.
Abstract: The capacity of ad-hoc wireless networks is constrained by the mutual interference of concurrent transmissions between nodes. We study a model of an ad-hoc network where n nodes communicate in random source-destination pairs. These nodes are assumed to be mobile. We examine the per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery. Under this assumption, the per-user throughput can increase dramatically when the nodes are mobile rather than fixed. This improvement can be achieved by exploiting node mobility as a type of multiuser diversity.
1,376 citations
Cites background from "Stable non-Gaussian random processe..."
...We have evaluated the asymptotic fraction of feasible pairs for the special case , because for this case, the normalized interference has L´ evy distribution2 [ 7 ], with...
[...]
...for general ; only the Laplace transform of its density is known explicitly [2, 7 ], and is given by . 3 This can be seen by comparing the Laplace transform of the den-...
[...]
TL;DR: A number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed, including Ornstein-Uhlenbeck type processes, superpositions of such processes and Stochastic volatility models in one and more dimensions.
Abstract: With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.
1,323 citations
Cites background from "Stable non-Gaussian random processe..."
...Of most interest, both theoretically and practically, among such processes are those for which the increments are stationary, and examples of these are generally difficult to come by, see for instance Samorodnitsky and Taqqu (1994) and references given there....
[...]
References
More filters
TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
7,412 citations
TL;DR: The per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery, can be increased dramatically under this assumption, and a form of multiuser diversity via packet relaying is exploited.
Abstract: The capacity of ad hoc wireless networks is constrained by the mutual interference of concurrent transmissions between nodes. We study a model of an ad hoc network where n nodes communicate in random source-destination pairs. These nodes are assumed to be mobile. We examine the per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery. Under this assumption, the per-user throughput can increase dramatically when nodes are mobile rather than fixed. This improvement can be achieved by exploiting a form of multiuser diversity via packet relaying.
2,736 citations
Book•
12 Jun 2007TL;DR: Random Fields and Geometry as discussed by the authors is a comprehensive survey of the general theory of Gaussian random fields with a focus on geometric problems arising in the study of random fields, including continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities.
Abstract: * Recasts topics in random fields by following a completely new way of handling both geometry and probability
* Significant exposition of the work of others in the field
* Presentation is clear and pedagogical
* Excellent reference work as well as excellent work for self study
This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined.
The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities.
"Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.
1,465 citations
22 Apr 2001
TL;DR: The per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery, can be increased dramatically when the nodes are mobile rather than fixed, by exploiting node mobility as a type of multiuser diversity.
Abstract: The capacity of ad-hoc wireless networks is constrained by the mutual interference of concurrent transmissions between nodes. We study a model of an ad-hoc network where n nodes communicate in random source-destination pairs. These nodes are assumed to be mobile. We examine the per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery. Under this assumption, the per-user throughput can increase dramatically when the nodes are mobile rather than fixed. This improvement can be achieved by exploiting node mobility as a type of multiuser diversity.
1,376 citations
TL;DR: A number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed, including Ornstein-Uhlenbeck type processes, superpositions of such processes and Stochastic volatility models in one and more dimensions.
Abstract: With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.
1,323 citations