Random Variables and Probability DistributionsCambridge University Press, 3 juin 2004 - 132 pages This 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. |
Table des matières
Axioms and preliminary theorems | 9 |
General properties Mean values | 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 | |
Expressions et termes fréquents
a₁ absolutely continuous absolutely convergent asymptotic expansion axioms Borel set Central Limit Theorem Chapter characteristic functions coefficients consider constant continuity point converges i.pr corresponding c.f. denote discontinuities equal components equal to zero F₁ F₂ finite number finite t-interval Fn(x following theorem function F given hypothesis independent random variables independent variables inequality integral Kolmogoroff Lemma Let X1 Lévy Liapounoff lim sup Lindeberg condition logf m₁ M₂ mathematical mutually independent never decreasing function non-negative normal distribution number of dimensions obtain obviously one-dimensional order moments P₁ P₂ particular Poisson distribution polynomial pr.f probability distribution proof of Theorem relation s.i.i. process S₁ S₂ satisfied second member semi-invariants sequence space tends to zero Theorem 11 Theorem 20 theory tw+1 uniformly uniquely determined variable X₁ X₂