Random Series and Stochastic Integrals: Single and Multiple

Single and Multiple
1992
ISBN: 978-0-8176-4198-6
Verlag: Birkhäuser Boston

Buch, Englisch, 360 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 1170 g

Reihe: Probability and Its Applications

This book studies the foundations of the theory of linear and nonlinear forms in single and multiple random variables including the single and multiple random series and stochastic integrals, both Gaussian and non-Gaussian. This subject is intimately connected with a number of classical problems of probability theory such as the summation of independent random variables, martingale theory, and Wiener's theory of polynomial chaos. The book contains a number of older results as well as more recent, or previously unpublished, results. The emphasis is on domination principles for comparison of different sequences of random variables and on decoupling techniques. These tools prove very useful in many areas ofprobability and analysis, and the book contains only their selected applications. On the other hand, the use of the Fourier transform - another classical, but limiting, tool in probability theory - has been practically eliminated. The book is addressed to researchers and graduate students in prob­ ability theory, stochastic processes and theoretical statistics, as well as in several areas oftheoretical physics and engineering. Although the ex­ position is conducted - as much as is possible - for random variables with values in general Banach spaces, we strive to avoid methods that would depend on the intricate geometric properties of normed spaces. As a result, it is possible to read the book in its entirety assuming that all the Banach spaces are simply finite dimensional Euclidean spaces.

Zielgruppe

Research

Autoren/Hrsg.

Woyczynski, Wojbor

Kwapien, Stanislaw

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Elementare Stochastik
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Naturwissenschaften Physik Physik Allgemein Geschichte der Physik
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Naturwissenschaften Physik Physik Allgemein Experimentalphysik
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

0 Preliminaries.- 0.1 Topology and measures.- 0.2 Tail inequalities.- 0.3 Filtrations and stopping times.- 0.4 Extensions of probability spaces.- 0.5 Bernoulli and canonical Gaussian and ?-stable sequences.- 0.6 Gaussian measures on linear spaces.- 0.7 Modulars on linear spaces.- 0.8 Musielak-Orlicz spaces.- 0.9 Random Musielak-Orlicz spaces.- 0.10 Complements and comments.- Bibliographical notes.- I Random Series.- 1 Basic Inequalities for Random Linear Forms in Independent Random Variables.- 2 Convergence of Series of Independent Random Variables.- 3 Domination Principles and Comparison of Sums of Independent Random Variables.- 4 Martingales.- 5 Domination Principles for Martingales.- 6 Random Multilinear Forms in Independent Random Variables and Polynomial Chaos.- II Stochastic Integrals.- 7 Integration with Respect to General Stochastic Measures.- 8 Deterministic Integrands.- 9 Predictable Integrands.- 10 Multiple Stochastic Integrals.- A Unconditional and Bounded Multiplier Convergence of Random Series.- A.2 Almost sure convergence.- A.3 Complements and comments.- A hypercontractive view.- Bibliographical notes.- B Vector Measures.- B.1 Extensions of vector measures.- B.2 Boundedness and control measure of stochastic measures.- B.3 Complements and comments.- Bibliographical notes.