Book contents
- Front matter
- CONTENTS
- Preface to the First Edition
- Preface to the Second Edition
- Preface to the Third Edition
- Abbreviations and Notations
- FIRST PART PRINCIPLES
- SECOND PART DISTRIBUTIONS IN R1
- Chapter III General properties. Mean values
- Chapter IV Characteristic functions
- Chapter V Addition of independent variables. Convergence “in probability” Special distributions
- Chapter VI The normal distribution and the central limit theorem
- Chapter VII Error estimation. Asymptotic expansions
- Chapter VIII A class of stochastic processes
- THIRD PART DISTRIBUTIONS IN RK
- Bibliography
- Some Recent Works on Mathematical Probability
Chapter VIII - A class of stochastic processes
Published online by Cambridge University Press: 22 September 2009
- Front matter
- CONTENTS
- Preface to the First Edition
- Preface to the Second Edition
- Preface to the Third Edition
- Abbreviations and Notations
- FIRST PART PRINCIPLES
- SECOND PART DISTRIBUTIONS IN R1
- Chapter III General properties. Mean values
- Chapter IV Characteristic functions
- Chapter V Addition of independent variables. Convergence “in probability” Special distributions
- Chapter VI The normal distribution and the central limit theorem
- Chapter VII Error estimation. Asymptotic expansions
- Chapter VIII A class of stochastic processes
- THIRD PART DISTRIBUTIONS IN RK
- Bibliography
- Some Recent Works on Mathematical Probability
Summary
1. In the preceding Chapters, we have been concerned with distributions of sums of the type Zn = X1 + … + Xn, where the Xr are independent random variables. Zn is then a variable depending on a discontinuous parameter n, and the passage from Zn to Zn+1 means that Zn receives the additive contribution Xn+1, so that we have Zn+1 = Zn + Xn+1 where Zn and Xn+1 are independent.
Consider now the formation of Zn by successive addition of the mutually independent contributions X1, X2, …, and let us assume that each addition of a new contribution takes a finite time δ. (In a concrete interpretation the Xr might e.g. be the gains of a certain player during a series of games, every game requiring the time δ, so that Zn is the total gain realized after n games, or after the time nδ.)
The sum Zn then arises after the time nδ, and the d.f. of Zn is thus the d.f. of the sum that has been formed during the time interval (0, nδ). Suppose now that we allow δ to tend to zero and n to tend to infinity, in such a way that nδ tends to a finite limit τ. It is conceivable that the distribution of Zn may then tend to a definite limit, which will depend on the continuous time parameterτ.
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- Random Variables and Probability Distributions , pp. 89 - 99Publisher: Cambridge University PressPrint publication year: 1970