## Counterexamples in ProbabilityCounterexamples (in the usual mathematical sense) are powerful tools of mathematical theory. In this book the author gives more than 250 drawn from the whole field of probability theory and stochastic processes. The counterexamples are selected for their interest and for the importance of the theory they illustrate. Each section starts with a summary of definitions and main results, followed by counterexamples ordered by content and difficulty. Full references and additional sources are given. |

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### Contents

CLASSES OF RANDOM EVENTS | 3 |

INDEPENDENCE OF RANDOM EVENTS | 12 |

DISTRIBUTION FUNCTIONS OF RANDOM | 25 |

Copyright | |

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### Common terms and phrases

absolutely continuous According answer arbitrary belongs called ch.f Clearly concerning conclude condition Consider construct contains convergence corresponding counterexamples defined definition Denote density dependent determined discrete easily equality equivalent ergodic Example exists expectation fact Feller filtration Finally finite fixed function Further given Hence hold implies independent r.v.s infinitely divisible integrable interesting interval introduce Kolmogorov limit martingale means measure moment problem moments Moreover mutually independent natural normal distribution Note o-field obeys obtain Obviously particular points Poisson positive possible present probability measure probability space problem properties question random variables random vector reader Recall relation respectively result satisfies sequence Shiryaev SLLN solution standard statement stationary Statistics stochastic differential equations stochastic processes strong sufficient Suppose symmetric theorem theory true uniformly uniquely values weak weakly Wiener process zero