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Basic Course in Probability Theory, A

Rabi Bhattacharya; Edward C. Waymire;


Institución detectada Año de publicación Navegá Descargá Solicitá
No detectada 2007 SpringerLink

Tabla de contenidos

Random Maps, Distribution, and Mathematical Expectation

In the spirit of a refresher, we begin with an overview of the measure-theoretic framework for probability. Readers for whom this is entirely new material may wish to consult the appendices for statements and proofs of basic theorems from analysis.

Palabras clave: Probability Space; Mathematical Expectation; Borel Measurable Function; Total Variation Distance; Cantelli Lemma.

Pp. 1-17

Independence, Conditional Expectation

Palabras clave: Independent Random Variable; Conditional Expectation; Product Measure; Closed Linear Subspace; Random Variable Versus.

Pp. 19-36

Martingales and Stopping Times

The notion of “martingale” has proven to be among the most powerful ideas to emerge in probability in the last century. In this section some basic foundations are presented. A more comprehensive treatment of the theory and its applications is provided in our text on stochastic processes.

Pp. 37-48

Classical Zero–One Laws, Laws of Large Numbers and Deviations

The term law has various meanings within probability. It is sometimes used synonymously with distribution of a random variable. However, it also may refer to an event or phenomenon that occurs in some predictable sense, as in a “law of averages.” The latter is the context of the present section.

Palabras clave: Fourth Moment; Nonnegative Random Variable; Cantelli Lemma; Tail Event; Classical Zero.

Pp. 49-58

Weak Convergence of Probability Measures

Pp. 59-72

Fourier Series, Fourier Transform, and Characteristic Functions

Palabras clave: Characteristic Function; FOURIER Series; Inversion Formula; Fourier Inversion; Continuity Theorem.

Pp. 73-98

Classical Central Limit Theorems

Palabras clave: Independent Random Variable; Triangular Array; Continuity Theorem; Lindeberg Condition; Independent Normal Random Variable.

Pp. 99-105

Laplace Transforms and Tauberian Theorem

Palabras clave: Laplace Transform; Inversion Formula; Bernstein Polynomial; TAUBERIAN Theorem; Renewal Equation.

Pp. 107-120

Random Series of Independent Summands

Pp. 121-127

Kolmogorov's Extension Theorem and Brownian Motion

Palabras clave: Brownian Motion; Polish Space; Extension Theorem; Coordinate Projection; Standard Brownian Motion.

Pp. 129-140

Brownian Motion: The LIL and Some Fine-Scale Properties

Pp. 141-146

Skorokhod Embedding and Donsker's Invariance Principle

Palabras clave: Brownian Motion; Random Walk; Conditional Distribution; Markov Property; Invariance Principle.

Pp. 147-165

A Historical Note on Brownian Motion

Pp. 167-170


Tipo: libros

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Springer Nature

País de edición

Reino Unido

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