## Basic Principles and Applications of Probability TheoryProbability theory arose originally in connection with games of chance and then for a long time it was used primarily to investigate the credibility of testimony of witnesses in the “ethical” sciences. Nevertheless, probability has become a very powerful mathematical tool in understanding those aspects of the world that cannot be described by deterministic laws. Probability has succeeded in ?nding strict determinate relationships where chance seemed to reign and so terming them “laws of chance” combining such contrasting - tions in the nomenclature appears to be quite justi?ed. This introductory chapter discusses such notions as determinism, chaos and randomness, p- dictibility and unpredictibility, some initial approaches to formalizing r- domness and it surveys certain problems that can be solved by probability theory. This will perhaps give one an idea to what extent the theory can - swer questions arising in speci?c random occurrences and the character of the answers provided by the theory. 1. 1 The Nature of Randomness The phrase “by chance” has no single meaning in ordinary language. For instance, it may mean unpremeditated, nonobligatory, unexpected, and so on. Its opposite sense is simpler: “not by chance” signi?es obliged to or bound to (happen). In philosophy, necessity counteracts randomness. Necessity signi?es conforming to law – it can be expressed by an exact law. The basic laws of mechanics, physics and astronomy can be formulated in terms of precise quantitativerelationswhichmustholdwithironcladnecessity. |

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### Contents

5 | |

Probability Space 2 1 Finite Probability Space 19 19 | 18 |

1 Independence of σAlgebras | 53 |

General Theory of Stochastic Processes | 93 |

Probability Basic Notions Structure Methods 1 | 98 |

Limit Theorems | 119 |

Historic and Bibliographic Comments | 139 |

Applied Probability | 191 |

Filtering 257 | 256 |

Historic and Bibliographic Comments | 273 |

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absolutely continuous assume Borel sets characteristic function conditional probability Consider continuous function converges countable cylinder sets defined denote density determined diffusion process distribution function elementary events equation ergodic exists finite finite-dimensional distributions follows formula given Hence homogeneous independent increments independent random variables inequality interval jump discontinuities Kolmogorov large numbers law of large Lemma limit theorems linear log2 mapping Markov chain Markov process martingale measurable space metric space modification nonnegative O-algebra observed occur parameter phase space positive probability measure probability space probability theory problem process with independent Proof random element random function random walk relative frequency respect right-continuous right-hand side sample satisfies sequence of independent stationary sequence stochastic processes stochastically continuous stopping subset sufficient Suppose transition probability uniformly values Wiener process zero