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Basic Course in Probability Theory, A
Rabi Bhattacharya; Edward C. Waymire;
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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.
Independence, Conditional Expectation
Palabras clave: Independent Random Variable; Conditional Expectation; Product Measure; Closed Linear Subspace; Random Variable Versus.
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.
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.
Weak Convergence of Probability Measures
Fourier Series, Fourier Transform, and Characteristic Functions
Palabras clave: Characteristic Function; FOURIER Series; Inversion Formula; Fourier Inversion; Continuity Theorem.
Classical Central Limit Theorems
Palabras clave: Independent Random Variable; Triangular Array; Continuity Theorem; Lindeberg Condition; Independent Normal Random Variable.
Laplace Transforms and Tauberian Theorem
Palabras clave: Laplace Transform; Inversion Formula; Bernstein Polynomial; TAUBERIAN Theorem; Renewal Equation.
Random Series of Independent Summands
Kolmogorov's Extension Theorem and Brownian Motion
Palabras clave: Brownian Motion; Polish Space; Extension Theorem; Coordinate Projection; Standard Brownian Motion.
Brownian Motion: The LIL and Some Fine-Scale Properties
Skorokhod Embedding and Donsker's Invariance Principle
Palabras clave: Brownian Motion; Random Walk; Conditional Distribution; Markov Property; Invariance Principle.
A Historical Note on Brownian Motion