Author

# A. J. Stam

Bio: A. J. Stam is an academic researcher from University of Groningen. The author has contributed to research in topics: Random walk & Central limit theorem. The author has an hindex of 10, co-authored 31 publications receiving 1083 citations.

##### Papers

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TL;DR: A certain analogy is found to exist between a special case of Fisher's quantity of information I and the inverse of the “entropy power” of Shannon and this constitutes a sharpening of the uncertainty relation of quantum mechanics for canonically conjugated variables.

Abstract: A certain analogy is found to exist between a special case of Fisher's quantity of information I and the inverse of the “entropy power” of Shannon (1949, p. 60). This can be inferred from two facts: (1) Both quantities satisfy inequalities that bear a certain resemblance to each other. (2) There is an inequality connecting the two quantities. This last result constitutes a sharpening of the uncertainty relation of quantum mechanics for canonically conjugated variables. Two of these relations are used to give a direct proof of an inequality of Shannon (1949, p. 63, Theorem 15). Proofs are not elaborated fully. Details will be given in a doctoral thesis that is in preparation.

792 citations

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TL;DR: In this paper, it was shown that if X = (X 1, · · ··, Xn ) has uniform distribution on the sphere or ball in Ω with radius a, then the joint distribution of, ···, k, converges in total variation to the standard normal distribution on ℝ.

Abstract: If X = (X 1, · ··, Xn ) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.

103 citations

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TL;DR: In this paper, the authors considered renewal moments of negative order on the real line and provided necessary and sufficient conditions for α and γ to hold for any ρ > 1 and α ≦ ρ ≦ 1.

Abstract: Let F be a probability measure on the real line and G = Σ C(k)Fk ∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V(x) vary regularly of order p for x→ ∞ and either (1) V(x) F[x, ∞)→ α ≧ 0 or (2) V(x) C[x, ∞)→ γ ≧ 0. If ρ > 1 and F(–∞, 0) = 0, necessary and sufficient in order that V(x) G[x, ∞)→b, is that both (1) and (2) hold for suitable α and γ. For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.

56 citations

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TL;DR: This representations is used to prove asymptotic results about the random partition for n → ∞ about a stochastic number of cells.

39 citations

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TL;DR: In this paper, a bound on the Kullback-Leibler information distance between the distributions of the sample compositions without and with replacement is derived, which depends only on n, N, H.

Abstract: Summary Two random samples of size n are taken from a set containing N objects of H types, first with and then without replacement. Let d be the absolute (L1-)distance and I the Kullback-Leibler information distance between the distributions of the sample compositions without and with replacement. Sample composition is meant with respect to types; it does not matter whether order of sampling is included or not. A bound on I and d is derived, that depends only on n, N, H. The bound on I is not higher than 2I. For fixed H we have d0, I0 as N if and only if n/N0. Let Wr be the epoch at which for the r-th time an object of type I appears. Bounds on the distances between the joint distributions of W1., Wr without and with replacement are given.

36 citations

##### Cited by

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TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

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02 Jan 2013

TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.

Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

5,524 citations

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16 Jan 2012TL;DR: In this article, a comprehensive treatment of network information theory and its applications is provided, which provides the first unified coverage of both classical and recent results, including successive cancellation and superposition coding, MIMO wireless communication, network coding and cooperative relaying.

Abstract: This comprehensive treatment of network information theory and its applications provides the first unified coverage of both classical and recent results. With an approach that balances the introduction of new models and new coding techniques, readers are guided through Shannon's point-to-point information theory, single-hop networks, multihop networks, and extensions to distributed computing, secrecy, wireless communication, and networking. Elementary mathematical tools and techniques are used throughout, requiring only basic knowledge of probability, whilst unified proofs of coding theorems are based on a few simple lemmas, making the text accessible to newcomers. Key topics covered include successive cancellation and superposition coding, MIMO wireless communication, network coding, and cooperative relaying. Also covered are feedback and interactive communication, capacity approximations and scaling laws, and asynchronous and random access channels. This book is ideal for use in the classroom, for self-study, and as a reference for researchers and engineers in industry and academia.

2,442 citations

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TL;DR: The relevant theory which may also be used in the wider context of Operation Research is reviewed, various applications from the field of insurance and finance are discussed and an extensive list of references are guided towards further material.

Abstract: Extremal events play an increasingly important role in stochastic modelling in insurance and finance. Over many years, probabilists and statisticians have developed techniques for the description, analysis and prediction of such events. In the present paper, we review the relevant theory which may also be used in the wider context of Operation Research. Various applications from the field of insurance and finance are discussed. Via an extensive list of references, the reader is guided towards further material related to the above problem areas.

1,927 citations

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01 Jan 2006

TL;DR: In this paper, the Brownian forest and the additive coalescent were constructed for random walks and random forests, respectively, and the Bessel process was used for random mappings.

Abstract: Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

1,371 citations