Random Series and Stochastic Integrals: Single and Multiple

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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 chara