ProbabilityIn the Preface to the first edition, originally published in 1980, we mentioned that this book was based on the author's lectures in the Department of Mechanics and Mathematics of the Lomonosov University in Moscow, which were issued, in part, in mimeographed form under the title "Probabil ity, Statistics, and Stochastic Processors, I, II" and published by that Univer sity. Our original intention in writing the first edition of this book was to divide the contents into three parts: probability, mathematical statistics, and theory of stochastic processes, which corresponds to an outline of a three semester course of lectures for university students of mathematics. However, in the course of preparing the book, it turned out to be impossible to realize this intention completely, since a full exposition would have required too much space. In this connection, we stated in the Preface to the first edition that only probability theory and the theory of random processes with discrete time were really adequately presented. Essentially all of the first edition is reproduced in this second edition. Changes and corrections are, as a rule, editorial, taking into account com ments made by both Russian and foreign readers of the Russian original and ofthe English and Germantranslations [Sll]. The author is grateful to all of these readers for their attention, advice, and helpful criticisms. In this second English edition, new material also has been added, as follows: in Chapter 111, §5, §§7-12; in Chapter IV, §5; in Chapter VII, §§8-10. |
Contents
Introduction | 1 |
2 Algebras and oalgebras Measurable Spaces | 2 |
3 Methods of Introducing Probability Measures on Measurable Spaces 4 Random Variables I | 3 |
5 Random Elements | 5 |
6 Lebesgue Integral Expectation | 6 |
Outcomes 6 The Bernoulli Scheme II Limit Theorems Local 55324 | 17 |
3 Conditional Probability Independence | 23 |
4 Random Variables and Their Properties | 32 |
5 Rapidity of Convergence in the Strong Law of Large Numbers and in the Probabilities of Large Deviations 308 308 | 400 |
CHAPTER V | 404 |
2 Ergodicity and Mixing | 407 |
3 Ergodic Theorems | 409 |
CHAPTER VI | 415 |
2 Orthogonal Stochastic Measures and Stochastic Integrals | 423 |
3 Spectral Representation of Stationary Wide Sense Sequences | 429 |
4 Statistical Estimation of the Covariance Function and the Spectral Density | 440 |
5 The Bernoulli Scheme I The Law of Large Numbers | 45 |
De MoivreLaplace Poisson | 55 |
7 Estimating the Probability of Success in the Bernoulli Scheme 32353 55 | 70 |
8 Conditional Probabilities and Mathematical Expectations with | 76 |
9 Random Walk I Probabilities of Ruin and Mean Duration in | 83 |
10 Random Walk II Reflection Principle Arcsine Law | 94 |
11 Martingales Some Applications to the Random Walk 94 | 103 |
12 Markov Chains Ergodic Theorem Strong Markov Property | 110 |
CHAPTER II | 131 |
7 Conditional Probabilities and Conditional Expectations with | 180 |
9 Construction of a Process with Given FiniteDimensional Distribution | 297 |
10 Various Kinds of Convergence of Sequences of Random Variables 11 The Hilbert Space of Random Variables with Finite Second Moment 12 Ch... | 301 |
CHAPTER III | 308 |
2 Relative Compactness and Tightness of Families of Probability Distributions | 317 |
3 Proofs of Limit Theorems by the Method of Characteristic Functions | 321 |
4 Central Limit Theorem for Sums of Independent Random Variables I | 328 |
5 Central Limit Theorem for Sums of Independent Random Variables II | 337 |
6 Infinitely Divisible and Stable Distributions | 341 |
7 Metrizability of Weak Convergence | 348 |
8 On the Connection of Weak Convergence of Measures with Almost Sure Convergence of Random Elements Method of a Single Probability Space | 353 |
9 The Distance in Variation between Probability Measures KakutaniHellinger Distance and Hellinger Integrals Application to Absolute Continuity an... | 359 |
10 Contiguity and Entire Asymptotic Separation of Probability Measures | 368 |
11 Rapidity of Convergence in the Central Limit Theorem | 373 |
12 Rapidity of Convergence in Poissons Theorem | 376 |
CHAPTER IV | 379 |
2 Convergence of Series | 384 |
3 Strong Law of Large Num bers 4 Law of the Iterated Logarithm | 395 |
5 Wolds Expansion | 446 |
6 Extrapolation Interpolation and Filtering | 453 |
7 The KalmanBucy Filter and Its Generalizations | 464 |
CHAPTER VII | 474 |
2 Preservation of the Martingale Property Under Time Change at a Random Time | 484 |
3 Fundamental Inequalities | 492 |
4 General Theorems on the Convergence of Submartingales and Martingales | 508 |
5 Sets of Convergence of Submartingales and Martingales | 515 |
6 Absolute Continuity and Singularity of Probability Distributions | 524 |
7 Asymptotics of the Probability of the Outcome of a Random Walk with Curvilinear Boundary | 536 |
8 Central Limit Theorem for Sums of Dependent Random Variables | 541 |
9 Discrete Version of Itos Formula | 554 |
10 Applications to Calculations of the Probability of Ruin in Insurance | 558 |
CHAPTER VIII | 564 |
2 Classification of the States of a Markov Chain in Terms of Arithmetic Properties of the Transition Probabilities p n | 569 |
3 Classification of the States of a Markov Chain in Terms | 571 |
Asymptotic Properties of the Probabilities pi n | 573 |
4 On the Existence of Limits and of Stationary Distributions | 582 |
5 Examples | 587 |
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Common terms and phrases
a₁ algebra B₁ B₂ Bernoulli random variables Bernoulli scheme Borel Chapter characteristic function Chebyshev's inequality completes the proof conditional expectation Consequently consider convergence Corollary countable D₁ decomposition defined Definition denote dF(x distributed random variables distribution function equation ergodic establishes estimator event example finite follows formula Gaussian Hence hypothesis identically distributed random independent identically distributed independent random variables inequality integral large numbers law of large Lebesgue measure lemma Let us show limit theorem linear Markov chain Markov property martingale matrix n₁ nonnegative o-algebra orthogonal P₁ particle probabilistic probability measure probability space probability theory Problem prove recurrent respect S₁ satisfied sequence of independent spectral density stationary sequence stochastic measure stochastic sequence submartingale subsets suppose values vector X₁ zero Εξ