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Über dieses Buch

This text develops the necessary background in probability theory underlying diverse treatments of stochastic processes and their wide-ranging applications. In this second edition, the text has been reorganized for didactic purposes, new exercises have been added and basic theory has been expanded. General Markov dependent sequences and their convergence to equilibrium is the subject of an entirely new chapter. The introduction of conditional expectation and conditional probability very early in the text maintains the pedagogic innovation of the first edition; conditional expectation is illustrated in detail in the context of an expanded treatment of martingales, the Markov property, and the strong Markov property. Weak convergence of probabilities on metric spaces and Brownian motion are two topics to highlight. A selection of large deviation and/or concentration inequalities ranging from those of Chebyshev, Cramer–Chernoff, Bahadur–Rao, to Hoeffding have been added, with illustrative comparisons of their use in practice. This also includes a treatment of the Berry–Esseen error estimate in the central limit theorem.
The authors assume mathematical maturity at a graduate level; otherwise the book is suitable for students with varying levels of background in analysis and measure theory. For the reader who needs refreshers, theorems from analysis and measure theory used in the main text are provided in comprehensive appendices, along with their proofs, for ease of reference.
Rabi Bhattacharya is Professor of Mathematics at the University of Arizona. Edward Waymire is Professor of Mathematics at Oregon State University. Both authors have co-authored numerous books, including a series of four upcoming graduate textbooks in stochastic processes with applications.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Random Maps, Distribution, and Mathematical Expectation

In the spirit of a refresher, we begin with an overview of the measure–theoretic framework for probability.
Rabi Bhattacharya, Edward C. Waymire

Chapter 2. Independence, Conditional Expectation

The notions of statistical independence, conditional expectation and conditional probability are the cornerstones of probability theory.
Rabi Bhattacharya, Edward C. Waymire

Chapter 3. Martingales and Stopping Times

The notion of martingale has proven to be among the most powerful ideas to emerge in probability in the past century.
Rabi Bhattacharya, Edward C. Waymire

Chapter 4. Classical Central Limit Theorems

In view of the great importance of the central limit theorem (CLT), we shall give a general but self-contained version due to Lindeberg.
Rabi Bhattacharya, Edward C. Waymire

Chapter 5. Classical Zero–One Laws, Laws of Large Numbers and Large Deviations

The term law has various meanings within probability.
Rabi Bhattacharya, Edward C. Waymire

Chapter 6. Fourier Series, Fourier Transform, and Characteristic Functions

Fourier series and Fourier transform provide one of the most important tools for analysis and partial differential equations, with widespread applications to physics in particular and science in general. This is (up to a scalar multiple) a norm-preserving (i.e., isometry), linear transformation on the Hilbert space of square-integrable complex-valued functions. It turns the integral operation of convolution of functions into the elementary algebraic operation of the product of the transformed functions, and that of differentiation of a function into multiplication by its Fourier frequency.
Rabi Bhattacharya, Edward C. Waymire

Chapter 7. Weak Convergence of Probability Measures on Metric Spaces

Let \((S,\rho )\) be a metric space and let \(\mathcal {P}(S)\) be the set of all probability measures on \((S, \mathcal {B}(S)).\) In this chapter we consider a general formulation of convergence in \(\mathcal {P}(S)\), referred to as weak convergence or convergence in distribution.
Rabi Bhattacharya, Edward C. Waymire

Chapter 8. Random Series of Independent Summands

The convergence of an infinite series \(\sum _{n=1}^\infty X_n\) is a tail event.
Rabi Bhattacharya, Edward C. Waymire

Chapter 9. Kolmogorov’s Extension Theorem and Brownian Motion

Suppose a probability measure Q is given on a product space \(\varOmega = \mathop {\prod }_{t\in \varLambda } S_{t}\) with the product \(\sigma \)-field \(\mathcal {F}= \otimes _{t \in \varLambda } \mathcal {S}_t\).
Rabi Bhattacharya, Edward C. Waymire

Chapter 10. Brownian Motion: The LIL and Some Fine-Scale Properties

In this chapter, we analyze the growth of the Brownian paths \(t\mapsto B_t\) as \(t\rightarrow \infty \).
Rabi Bhattacharya, Edward C. Waymire

Chapter 11. Strong Markov Property, Skorokhod Embedding, and Donsker’s Invariance Principle

This chapter ties together a number of the topics introduced in the text via applications to the further analysis of Brownian motion, a fundamentally important stochastic process whose existence was established in Chapter IX.
Rabi Bhattacharya, Edward C. Waymire

Chapter 12. A Historical Note on Brownian Motion

Historically, the mathematical roots of Brownian motion lie in the central limit theorem (CLT).
Rabi Bhattacharya, Edward C. Waymire

Chapter 13. Some Elements of the Theory of Markov Processes and Their Convergence to Equilibrium

Special examples of Markov processes, such as random walks in discrete time and Brownian motion in continuous time, have occurred many times in preceding chapters as illustrative examples of martingales and Markov processes.
Rabi Bhattacharya, Edward C. Waymire

Backmatter

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