## Random Variables and Probability DistributionsThis tract develops the purely mathematical side of the theory of probability, without reference to any applications. When originally published, it was one of the earliest works in the field built on the axiomatic foundations introduced by A. Kolmogoroff in his book Grundbegriffe der Wahrscheinlichkeitsrechnung, thus treating the subject as a branch of the theory of completely additive set functions. The author restricts himself to a consideration of probability distributions in spaces of a finite number of dimensions, and to problems connected with the Central Limit Theorem and some of its generalizations and modifications. In this edition the chapter on Liapounoff's theorem has been partly rewritten, and now includes a proof of the important inequality due to Berry and Esseen. The terminology has been modernized, and several minor changes have been made. |

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### Contents

Axioms and preliminary theorems | 9 |

SECOND PART DISTRIBUTIONS IN | 18 |

Characteristic functions | 24 |

Addition of independent variables Conver | 36 |

The normal distribution and the central limit | 50 |

Error estimation Asymptotic expansions | 70 |

A class of stochastic processes | 89 |

THIRD PART DISTRIBUTIONS IN | 100 |

The normal distribution and the central limit | 109 |

115 | |

### Common terms and phrases

absolutely continuous absolutely convergent assumes the value asymptotic expansion axioms Bernoulli distribution Borel set Central Limit Theorem Chapter characteristic function coefficients combined variable condition 64 consider constant continuity point corresponding c.f. denote dimensional discontinuities distribution in Rk equal components equal to zero finite interval finite mean value finite number following theorem given hypothesis independent random variables independent variables inequality integral Kolmogoroff Lebesgue-Stieltjes integral Lemma Liapounoff lim sup Lindeberg condition mathematical mutually independent never decreasing function non-negative normal distribution number of dimensions obtain obviously one-dimensional order to prove parameter particular Poisson distribution polynomial pr.f probability distribution proof of Theorem relation s.i.i. process second member semi-invariants space sum Xl tends to zero Theorem 11 Theorem 20 Theorem 9 theory uniformly uniquely determined variable X1 variable ZT variables in Rk vergence Xlt X2 Xv X2