Random Series and Stochastic Integrals: Single and Multiple: Single and Multiple

ISBN-13: 9780817641986 / Angielski / Miękka / 2000 / 360 str.

Stanislaw Kwapien; Wojbor Woyczynski

Opis

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 The book contains a number of older results as well as more recent, or previously unpublished, results.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.

Spis treści

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.- 1.1 Lévy-Octaviani inequalities.- 1.2 Contraction inequalities.- 1.3 Moment inequalities.- 1.4 Complements and comments.- Best constants in the Lévy-Octaviani inequality.- A contraction inequality for mixtures of Gaussian random variables.- Tail inequalities for Bernoulli and Gaussian random linear forms.- A refinement of the moment inequality.- Comparison of moments.- Bibliographical notes.- 2 Convergence of Series of Independent Random Variables.- 2.1 The Itô-Nisio Theorem.- 2.2 Convergence in the p-th mean.- 2.3 Exponential and other moments of random series.- 2.4 Random series in function spaces.- 2.5 An example: construction of the Brownian motion.- 2.6 Karhunen-Loève representation of Gaussian measures.- 2.7 Complements and comments.- Rosenthal’s inequalities.- Strong exponential moments of Gaussian series.- Lattice function spaces.- Convergence of Gaussian series.- Bibliographical notes.- 3 Domination Principles and Comparison of Sums of Independent Random Variables.- 3.1 Weak domination.- 3.2 Strong domination.- 3.3 Hypercontractive domination.- 3.4 Hypercontractivity of Bernoulli and Gaussian series.- 3.5 Sharp estimates of growth of p-th moments.- 3.6 Complements and comments.- More on C-domination.- Superstrong domination.- Domination of character systems on compact Abelian groups.- Random matrices.- Hypercontractivity of real random variables.- More precise estimates on strong exponential moments of Gaussian series.- Growth of p-th moments revisited.- More on strong exponential moments of series of bounded variables.- Bibliographical notes.- 4 Martingales.- 4.1 Doob’s inequalities.- 4.2 Convergence of martingales.- 4.3 Tangent and decoupled sequences.- 4.4 Complements and comments.- Bibliographical notes.- 5 Domination Principles for Martingales.- 5.1 Weak domination.- 5.2 Strong domination.- 5.3 Burkholder’s method: comparison of subordinated martingales.- 5.4 Comparison of strongly dominated martingales.- 5.5 Gaussian martingales.- 5.6 Classic martingale inequalities.- 5.7 Comparison of the a.s convergence of series of tangent sequences.- 5.8 Complements and comments.- Tangency and ergodic theorems.- Burkholder’s method for conditionally Gaussian and conditionally independent martingales.- Necessity of moderate growth of ?.- Comparison of Gaussian martingales revisited.- Comparing H-valued martingales with 2-D martingales.- The principle of conditioning in limit theorems.- Bibliographical notes.- 6 Random Multilinear Forms in Independent Random Variables and Polynomial Chaos.- 6.1 Basic definitions and properties.- 6.2 Maximal inequalities.- 6.3 Contraction inequalities and domination of polynomial chaos.- 6.4 Decoupling inequalities.- 6.5 Comparison of moments of polynomial chaos.- 6.6 Convergence of polynomial chaos.- 6.7 Quadratic chaos.- 6.8 Wiener chaos and Herrnite polynomials.- 6.9 Complements and comments.- Tail and moment comparisons for chaos and its decoupled chaos.- Necessity of the symmetry condition in decoupling inequalities.- Karhunen-Loève expansion for the Wiener chaos.- ?-stable chaos of degree d ? 2.- Bibliographical notes.- II Stochastic Integrals.- 7 Integration with Respect to General Stochastic Measures.- 7.1 Construction of the integral.- 7.2 Examples of stochastic measures.- 7.3 Complements and comments.- An alternative definition of m-integrability.- Bibliographical notes.- 8 Deterministic Integrands.- 8.1 Discrete stochastic measure.- 8.2 Processes with independent increments and their characteristics.- 8.3 Integration with respect to a general independently scattered measure.- 8.4 Complements and comments.- Stochastic measures with finite p-th moments.- Bibliographical notes.- 9 Predictable Integrands.- 9.1 Integration with respect to processes with independent increments: Decoupling inequalities approach.- 9.2 Brownian integrals.- 9.3 Characteristics of semimartingales.- 9.4 Semimartingale integrals.- 9.5 Complements and comments.- The Bichteler-Dellacherie Theorem.- Semimartingale integrals in Lp.- ?-stable integrals.- Bibliographical notes.- 10 Multiple Stochastic Integrals.- 10.1 Products of stochastic measures.- 10.2 Structure of double integrals.- 10.3 Wiener polynomial chaos revisited.- 10.4 Complements and comments.- Multiple ?-stable integrals.- Bibliographical notes.- 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.