Random Series and Stochastic Integrals: Single and Multiple (Probability and Its Applications) [2 ed.] 0817635726, 9780817635725

Table of contents :
Cover
Title
ISBN
Contents
Introduction
Notation
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 a-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
1 RANDOM SERIES
1 Basic Inequalities for Random Linear Forms in Independent Random Variables
1.1 Levy-Octaviani inequalities
1.2 Contraction inequalities
1.3 Moment inequalities
1.4 Complements and comments
Best constants in the Levy-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 Ito-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-Loeve 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 (p
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 Hermite 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-Loeve expansion for the Wiener chaos
a-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
a-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 a-stable integrals
Bibliographical notes
A Unconditional and Bounded Multiplier Convergence of Random Series
A.l Convergence in probability and in the p-th moment
A.2 Almost sure convergence
A.3 Complements and comments
A hypercontractive view
Bibliographical notes
B Vector Measures
B.l Extensions of vector measures
B.2 Boundedness and control measure of stochastic measures
B.3 Complements and comments
Bibliographical notes
Bibliography
Index