## Radically Elementary Probability TheoryUsing only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form. |

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### Contents

Algebras of random variables | 6 |

External concepts | 12 |

External analogues of internal notions | 20 |

The decomposition of a stochastic process | 33 |

Convergence of martingales | 41 |

Fluctuations of martingales | 48 |

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a.e. trajectory a.s. implies a.s. Proof absolutely continuous algebra of random arbitrary associated martingale atoms Borel set Chapter Chebyshev inequality conditional expectation constant random variable contains converges a.s. convex function Corollary discontinuity of f e-fluctuations element event external formula f is continuous finite probability space finite set finite subset fixed point function holds hypothesis independent random variables infinitely close infinitesimal interval Jensen's inequality large numbers law of large Let f Let tbe limited fluctuation a.s. limited number Lindeberg condition nearby elementary process nonstandard natural number normalized martingale Notice oo a.s. overspill P-process point of discontinuity Poisson walk probability theory properties prove quadratic variation real numbers relativization satisfies the Lindeberg sequence principle standard natural number standard stochastic process stochastic process indexed strong law submartingale supermartingale Suppose tbe a martingale tbe a stochastic Theorem 7.1 unlimited variables of mean Wiener process Wiener walk