Problems in probability

Cover......Page 1
Problems in Probability......Page 4
Preface......Page 6
Contents......Page 10
1.1 Probabilistic Models for Experiments with Finitely Many Outcomes......Page 14
1.2 Some Classical Models and Distributions......Page 26
1.3 Conditional Probability: Independence......Page 40
1.4 Random Variables and Their Characteristics......Page 42
1.5 Bernoulli Scheme I: The Law of Large Numbers......Page 47
1.6 Bernoulli Scheme II: Limit Theorems (Local, Moivre–Laplace, Poisson)......Page 50
1.7 Estimate of the Probability for Success in Bernoulli Trials......Page 53
1.8 Conditional Probabilities and Expectations with Respect to Partitions......Page 55
1.9 Random Walk I: Probability of Ruin and Time Until Ruin in Coin Tossing......Page 59
1.10 Random Walk II: The Reflection Principle and the Arcsine Law......Page 62
1.11 Martingales: Some Applications to the Random Walk......Page 67
1.12 Markov Chains: The Ergodic Theorem: The Strong Markov Property......Page 69
2.1 Probabilistic Models of Experiments with Infinitely Many Outcomes: Kolmogorov's Axioms......Page 72
2.2 Algebras and -algebras: Measurable Spaces......Page 77
2.3 Methods for Constructing Probability Measures on Measurable Spaces......Page 83
2.4 Random Variables I......Page 92
2.5 Random Elements......Page 96
2.6 The Lebesgue Integral: Expectation......Page 98
2.7 Conditional Probabilities and Conditional Expectations with Respect to -algebras......Page 119
2.8 Random Variables II......Page 127
2.9 Construction of Stochastic Processes with a Given System of Finite-Dimensional Distributions......Page 151
2.10 Various Types of Convergence of Sequencesof Random Variables......Page 152
2.11 Hilbert Spaces of Random Variables with FiniteSecond Moments......Page 162
2.12 Characteristic Functions......Page 164
2.13 Gaussian Systems of Random Variables......Page 177
3.1 Weak Convergence of Probability Measures and Distributions......Page 193
3.2 Relative Compactness and Tightness of Families of Probability Distributions......Page 198
3.3 The Method of Characteristic Functions for EstablishingLimit Theorems......Page 200
3.4 The Central Limit Theorem for Sums of Independent Random Variables I. Lindeberg's Condition......Page 207
3.6 Infinitely Divisible and Stable Distributions......Page 217
3.7 Metrizability of the Weak Convergence......Page 223
3.8 The Connection Between Almost Sure Convergence and Weak Convergence of Probability Measures (the ``Common Probability Space'' Method)......Page 225
3.9 The Variation Distance Between Probability Measures. The Kakutani-Hellinger Distance and the Hellinger Integral. Applications to Absolute Continuity and Singularity of Probability Measures......Page 228
3.10 Contiguity (Proximity) and Full Asymptotic Separation of Probability Measures......Page 235
3.11 Rate of Convergence in the Central Limit Theorem......Page 237
3.12 Rate of Convergence in the Poisson Limit Theorem......Page 238
3.13 The Fundamental Theorems of Mathematical Statistics......Page 240
4.1 0–1 Laws......Page 245
4.2 Convergence of Series of Random Variables......Page 249
4.3 The Strong Law of Large Numbers......Page 255
4.4 The Law of the Iterated Logarithm......Page 261
4.5 Rate of Convergence in the Strong Law of Large Numbers......Page 265
5.1 Stationary (in Strict Sense) Random Sequences: Measure-Preserving Transformations......Page 270
5.2 Ergodicity and Mixing......Page 272
5.3 Ergodic Theorems......Page 273
6.1 Spectral Representation of Covariance Functions......Page 278
6.2 Orthogonal Stochastic Measures and Stochastic Integrals......Page 280
6.3 Spectral Representations of Stationary (in Broad Sense)Sequences......Page 281
6.4 Statistical Estimates of Covariance Functions and Spectral Densities......Page 283
6.5 Wold Decomposition......Page 285
6.6 Extrapolation, Interpolation and Filtartion......Page 286
6.7 The Kalman-Bucy Filter and Its Generalizations......Page 288
7.1 The Notion of Martingale and Related Concepts......Page 293
7.2 Invariance of the Martingale Property Under Random Time-Change......Page 298
7.3 Fundamental Inequalities......Page 304
7.4 Convergence Theorems for Submartingales and Martingales......Page 314
7.5 On the Sets of Convergence of Submartingales and Martingales......Page 320
7.6 Absolute Continuity and Singularity of Probability Distributions on Measurable Spaces with Filtrations......Page 321
7.7 On the Asymptotics of the Probability for a Random Walk to Exit on a Curvilinear Boundary......Page 324
7.8 The Central Limit Theorem for Sums of Dependent Random Variables......Page 326
7.9 Discrete Version of the Itô Formula......Page 328
7.10 The Probability for Ruin in Insurance. Martingale Approach......Page 330
7.11 On the Fundamental Theorem of Financial Mathematics: Martingale Characterization of the Absence of Arbitrage......Page 333
7.12 Hedging of Financial Contracts in Arbitrage-Free Markets......Page 334
7.13 The Optimal Stopping Problem: Martingale Approach......Page 335
8.1 Definitions and Basic Properties......Page 338
8.2 The Strong and the Generalized Markov Properties......Page 342
8.4 Markov Chain State Classification Based on the Transition Probability Matrix......Page 347
8.5 Markov Chain State Classification Based on the Asymptotics of the Transition Probabilities......Page 348
8.6-7 On the Limiting, Stationary, and Ergodic Distributions of Markov Chains with at Most Countable State Space......Page 354
8.8 Simple Random Walks as Markov Chains......Page 356
8.9 The Optimal Stopping Problem for Markov Chains......Page 364
A.1 Elements of Combinatorics......Page 367
A.2 Basic Probabilistic Structures and Concepts......Page 373
A.3 Analytical Methods and Tools of Probability Theory......Page 377
A.4 Stationary (in Strict Sense) Random Sequences......Page 393
A.5 Stationary (in Broad Sense) Random Sequences......Page 394
A.6 Martingales......Page 395
A.7 Markov Chains......Page 396
References......Page 415
Author Index......Page 420
Subject Index......Page 423