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1988 | Buch | 2. Auflage

Classes of Sets, Measures, and Probability Spaces Kapitel lesen Erstes Kapitel lesen

Probability Theory

Independence, Interchangeability, Martingales

verfasst von: Yuan Shih Chow, Henry Teicher

Verlag: Springer US

Buchreihe : Springer Texts in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

Apart from new examples and exercises, some simplifications of proofs, minor improvements, and correction of typographical errors, the principal change from the first edition is the addition of section 9.5, dealing with the central limit theorem for martingales and more general stochastic arrays. vii Preface to the First Edition Probability theory is a branch of mathematics dealing with chance phenomena and has clearly discernible links with the real world. The origins of the sub­ ject, generally attributed to investigations by the renowned French mathe­ matician Fermat of problems posed by a gambling contemporary to Pascal, have been pushed back a century earlier to the Italian mathematicians Cardano and Tartaglia about 1570 (Ore, 1953). Results as significant as the Bernoulli weak law of large numbers appeared as early as 1713, although its counterpart, the Borel strong law oflarge numbers, did not emerge until 1909. Central limit theorems and conditional probabilities were already being investigated in the eighteenth century, but the first serious attempts to grapple with the logical foundations of probability seem to be Keynes (1921), von Mises (1928; 1931), and Kolmogorov (1933).

Inhaltsverzeichnis

Frontmatter
1. Classes of Sets, Measures, and Probability Spaces
Abstract
A set in the words of Georg Cantor, the founder of modern set theory, is a collection into a whole of definite, well-distinguished objects of our perception or thought, The objects are called elements and the set is the aggregate of these elements. It is very convenient to extend this notion and also envisage a set devoid of elements, a so-called empty set, and this will be denoted by Ø. Each element of a set appears only once therein and its order of appearance within the set is irrelevant. A set whose elements are themselves sets will be called a class.
Yuan Shih Chow, Henry Teicher
2. Binomial Random Variables
Abstract
The major theorems of probability theory fall into a natural dichotomythose which are analytic in character and those which are measure-theoretic. In the latter category are zero—one laws, the Borel—Cantelli lemma, strong laws oflarge numbers, and indeed any result which requires the apparatus of a probability space.
Yuan Shih Chow, Henry Teicher
3. Independence
Abstract
Independence may be considered the single most important concept in probability theory, demarcating the latter from measure theory and fostering an independent development. In the course of this evolution, probability theory has been fortified by its links with the real world, and indeed the definition of independence is the abstract counterpart of a highly intuitive and empirical notion. Independence of random variables {X i}, the definition of which involves the events ofσ(X i) will be shown in Section 2 to concern only the joint distribution functions.
Yuan Shih Chow, Henry Teicher
4. Integration in a Probability Space
Abstract
There are two basic avenues to integration. In the modern approach the integral is introduced first for simple functions—as a weighted average of the values of the function-and then defined for any nonnegative measurable function f as a limit of the integrals of simple nonnegative functions increasing to f Conceptually this is extremely simple, but a certain price is paid in terms of proofs. The alternative classical approach, while employing a less intuitive definition, achieves considerable simplicity in proofs of elementary properties.
Yuan Shih Chow, Henry Teicher
5. Sums of Independent Random Variables
Abstract
Of paramount concern in probability theory is the behavior of sums {S n , n ≥ 1} of independent random variables {X i, i ≥ 1}. The case where the {X i} are i.i.d. is of especial interest and frequently lends itself to more incisive results. The sequence of sums {S n, n ≥ 1} of i.i.d. r.v.s {X n} is alluded to as a random walk; in the particular case when the component r.v.s {X n} are nonnegative, the random walk is referred to as a renewal process.
Yuan Shih Chow, Henry Teicher
6. Measure Extensions, Lebesgue-Stieltjes Measure Kolmogorov Consistency Theorem
Abstract
A salient underpinning of probability theory is the one-to-one correspondence between distribution functions on R n and probability measures on the Borel subsets of R n . Verification of this correspondence involves the notion of measure extension.
Yuan Shih Chow, Henry Teicher
7. Conditional Expectation, Conditional Independence, Introduction to Martingales
Abstract
From a theoretical vantage point, conditioning is a useful means of exploiting auxiliary information. From a practical vantage point, conditional probabilities reflect the change in unconditional probabilities due to additional knowledge.
Yuan Shih Chow, Henry Teicher
8. Distribution Functions and Characteristic Functions
Abstract
Distribution functions are mathematical artifacts with properties that are independent of any probabilistic setting. Notwithstanding, most of the theorems of interest are geared to d.f.s of r.v.s and the majority of proofs are simpler and more intuitive when couched in terms of r.v.s having, or probability measures determined by, the given d.f.s. Since r.v.s possessing pre- assigned d.f.s can always be defined on some probability space, the language of r.v.s and probability will be utilized in many of the proofs without further ado.
Yuan Shih Chow, Henry Teicher
9. Central Limit Theorems
Abstract
Central limit theorems have played a paramount role in probability theory starting—in the case of independent random variables—with the DeMoivre- Laplace version and culminating with that of Lindeberg—Feller. The term “central” refers to the pervasive, although nonunique, role of the normal distribution as a limit of d.f.s of normalized sums of (classically independent) random variables. Central limit theorems also govern various classes of dependent random variables and the cases of martingales and interchangeable random variables will be considered.
Yuan Shih Chow, Henry Teicher
10. Limit Theorems for Independent Random Variables
Abstract
Prior discussion of the strong and weak laws of large numbers centered around the i.i.d. case. Necessary and sufficient conditions for the weak law are available when the underlying random variables are merely independent and have recently been obtained for the strong law as well. Unfortunately, the practicality of the latter conditions leaves much to be desired.
Yuan Shih Chow, Henry Teicher
11. Martingales
Abstract
An introduction to martingales appeared in Section 7.4, where convergence theorems for submartingales {S n , ℱ n , n ≥ 1} (relating to differentiation theory) were discussed. Here, emphasis will fall upon convergence theorems for martingales {S -n , ℱ -n , n ≤ -1} (relating to ergodic theorems). In demarcating the two cases, it is natural to refer to a martingale {S n , ℱ n , n ≥ 1} as an upward martingale and to allude to a martingale {S -n , ℱ -n , n ≤ -1} when written {S n , ℱ n , n ≥ 1} as a downward or reverse martingale. Martingale and stochastic inequalities will also be dealt with.
Yuan Shih Chow, Henry Teicher
12. Infinitely Divisible Laws
Abstract
It is a remarkable fact that the class of limit distributions of normed sums of i.i.d. random variables is severely circumscribed. If the underlying r.v.s, say {X n , n ≥ 1} have merely absolute moments of order r, then for r ≥ 2 only the normal distribution can arise as a limit, while if 0 < r ≤ 2, the limit law belongs to a class called stable distributions. If the basic r.v.s are merely independent (and infinitesimal when normed cf. (1) of Section 2), a larger class of limit laws, the so-called class ℒ emerges. But even the class ℒ does not contain a distribution of such crucial importance as the Poisson. A perusal of the derivation (Chapter 2) of the Poisson law as a limit of binomial laws B n reveals that the success probability associated with B n is a function of n. Thus, if B n-1 is envisaged as the distribution of the sum of i.i.d. random variables Y l, …, Y n-1, then B n must be the distribution of the sum of n different i.i.d. random variables which, therefore, may as well be labeled X n, 1, X n, 2,..., X n, n .
Yuan Shih Chow, Henry Teicher
Backmatter
Metadaten
Titel
Probability Theory
verfasst von
Yuan Shih Chow
Henry Teicher
Copyright-Jahr
1988
Verlag
Springer US
Electronic ISBN
978-1-4684-0504-0
Print ISBN
978-1-4684-0506-4
DOI
https://doi.org/10.1007/978-1-4684-0504-0