Harald Luschgy's Books

Stable Convergence and Stable Limit Theorems

By: Erich Häusler,Harald Luschgy

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Stable Convergence and Stable Limit Theorems

By: Erich Häusler,Harald Luschgy

The authors present a concise but complete exposition of the mathematical theory of stable convergence and give various applications in different areas of probability theory and mathematical statistics to illustrate the usefulness of this concept. Stable convergence holds in many limit theorems of probability theory and statistics – such as the classical central limit theorem – which are usually formulated in terms of convergence in distribution. Originated by Alfred Rényi, the notion of stable convergence is stronger than the classical weak convergence of probability measures. A variety of methods is described which can be used to establish this stronger stable convergence in many limit theorems which were originally formulated only in terms of weak convergence. Naturally, these stronger limit theorems have new and stronger consequences which should not be missed by neglecting the notion of stable convergence. The presentation will be accessible to researchers and advanced students at the master's level with a solid knowledge of measure theoretic probability.

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Page Count

231

Publisher

Springer

Published Date

ISBN 10

331918329X

ISBN 13

9783319183299

Foundations of Quantization for Probability Distributions

By: Siegfried Graf,Harald Luschgy

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Foundations of Quantization for Probability Distributions

By: Siegfried Graf,Harald Luschgy

Due to the rapidly increasing need for methods of data compression, quantization has become a flourishing field in signal and image processing and information theory. The same techniques are also used in statistics (cluster analysis), pattern recognition, and operations research (optimal location of service centers). The book gives the first mathematically rigorous account of the fundamental theory underlying these applications. The emphasis is on the asymptotics of quantization errors for absolutely continuous and special classes of singular probabilities (surface measures, self-similar measures) presenting some new results for the first time. Written for researchers and graduate students in probability theory the monograph is of potential interest to all people working in the disciplines mentioned above.

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Page Count

238

Publisher

Springer

Published Date

ISBN 10

3540455779

ISBN 13

9783540455776

Marginal and Functional Quantization of Stochastic Processes

By: Harald Luschgy,Gilles Pagès

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Marginal and Functional Quantization of Stochastic Processes

By: Harald Luschgy,Gilles Pagès

Vector Quantization, a pioneering discretization method based on nearest neighbor search, emerged in the 1950s primarily in signal processing, electrical engineering, and information theory. Later in the 1960s, it evolved into an automatic classification technique for generating prototypes of extensive datasets. In modern terms, it can be recognized as a seminal contribution to unsupervised learning through the k-means clustering algorithm in data science. In contrast, Functional Quantization, a more recent area of study dating back to the early 2000s, focuses on the quantization of continuous-time stochastic processes viewed as random vectors in Banach function spaces. This book distinguishes itself by delving into the quantization of random vectors with values in a Banach space—a unique feature of its content. Its main objectives are twofold: first, to offer a comprehensive and cohesive overview of the latest developments as well as several new results in optimal quantization theory, spanning both finite and infinite dimensions, building upon the advancements detailed in Graf and Luschgy's Lecture Notes volume. Secondly, it serves to demonstrate how optimal quantization can be employed as a space discretization method within probability theory and numerical probability, particularly in fields like quantitative finance. The main applications to numerical probability are the controlled approximation of regular and conditional expectations by quantization-based cubature formulas, with applications to time-space discretization of Markov processes, typically Brownian diffusions, by quantization trees. While primarily catering to mathematicians specializing in probability theory and numerical probability, this monograph also holds relevance for data scientists, electrical engineers involved in data transmission, and professionals in economics and logistics who are intrigued by optimal allocation problems.

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Page Count

918

Publisher

Springer Nature

Published Date

ISBN 10

3031454642

ISBN 13

9783031454646

Foundations of Quantization for Probability Distributions

By: Siegfried Graf,Harald Luschgy

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Foundations of Quantization for Probability Distributions

By: Siegfried Graf,Harald Luschgy

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Solve jigsaw puzzle

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Page Count

238

Publisher

Springer

Published Date

ISBN 10

3540455779

ISBN 13

9783540455776

Marginal and Functional Quantization of Stochastic Processes

By: Harald Luschgy,Gilles Pagès

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Marginal and Functional Quantization of Stochastic Processes

By: Harald Luschgy,Gilles Pagès

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Solve jigsaw puzzle

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Page Count

918

Publisher

Springer Nature

Published Date

ISBN 10

3031454642

ISBN 13

9783031454646

Book's by Harald Luschgy

Stable Convergence and Stable Limit Theorems

Stable Convergence and Stable Limit Theorems

Foundations of Quantization for Probability Distributions

Foundations of Quantization for Probability Distributions

Marginal and Functional Quantization of Stochastic Processes

Marginal and Functional Quantization of Stochastic Processes

Foundations of Quantization for Probability Distributions

Foundations of Quantization for Probability Distributions

Marginal and Functional Quantization of Stochastic Processes

Marginal and Functional Quantization of Stochastic Processes

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