## 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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additive algebra applied assume ball basis Borel bounded called characteristic function collection compact complete conditional Consider contains converges decision defined definition denote density depend determined diffusion discontinuities distribution function elements equals equation example exists expectation experiment expressed fact finite follows formula give given Hence holds implies increments independent integral interval jump Lemma limit linear mapping Markov process martingale means measure modification natural necessary O-algebra observed obtain occur operator positive possible probability measure probability space probability theory problem Proof prove random function random variables relation respect result right-hand side sample satisfies separable sequence solution specified statement stationary stochastic stopping subset sufficient Suppose Theorem transition probability uniformly values viewed walk Wiener zero