Probability TheoryProbability theory forms the basis of mathematical statistics, and has applications in many related areas. This comprehensive book tackles the principal problems and advanced questions of probability theory in 21 self-contained chapters, which are presented in logical order, but are also easy to deal with individually. The book is further distinguished by the inclusion of clear and illustrative proofs of the fundamental results. Probability theory is currently an extremely active area of research internationally, and the importance of the Russian school in the development of the subject has long been recognized. The frequent references to Russian literature throughout this work lend a fresh dimension to the book, and make it an invaluable source of reference for Western researchers and advanced students in probability related subjects. |
Contents
Introduction | 1 |
Arbitrary Space of Elementary Events | 14 |
Random Variables and Distribution Functions | 28 |
Numerical Characteristics of Random Variables | 56 |
Sequences of Independent Trials with Two Outcomes | 91 |
On Convergence of Random Variables and Distributions | 108 |
Characteristic Functions | 125 |
Sequences of Independent Random Variables Limit Theorems | 151 |
Information and Entropy | 284 |
Martingales | 293 |
Stationary Sequences | 325 |
Stochastically Recursive Sequences | 336 |
Continuous Time Random Processes | 352 |
Processes with Independent Increments | 362 |
Functional Limit Theorems | 377 |
Markov Processes | 392 |
Elements of Renewal Theory | 191 |
Sequences of Independent Random Variables Properties of the Trajectory | 209 |
Factorization Identities | 221 |
Sequences of Dependent Trials Markov Chains | 242 |
Processes with Finite Second Moments Gaussian Processes | 419 |
Appendices | 426 |
| 467 | |
Common terms and phrases
arbitrary assertion assume B₁ B₂ Bernoulli scheme Borel sets ch.f Chapter Chebyshev's inequality clearly coincide conditional expectation conditions of Theorem consider continuous convergence theorem Corollary countable defined Definition Denote density distributed random variables distribution function Ek,n equal equation ergodic event Example exists finite finite-dimensional distributions follows function g given hence holds implies independent random variables inequality integral interval k₁ large numbers law of large Lebesgue Lebesgue measure Lemma lim sup limit theorem Markov chain martingale means measure min{k o-algebra obtain particles Poisson process probability space Proof of Theorem prove random process relation respect right hand side satisfied Section sequence of independent stationary sequence submartingale sufficient total probability formula trajectory uniformly uniformly integrable values vector verify Wiener process Xn+1